Categorical-algebraic methods in non-commutative and non-associative algebra
Abstract
The objective of this dissertation is twofold: firstly to use categorical and algebraic methods to study homological properties of some of the aforementioned semi-abelian, non-associative structures and secondly to use categorical and algebraic methods to study categorical properties and provide categorical characterisations of some well-known algebraic structures. On one hand, the theory of universal central extensions together with the non-abelian tensor product will be studied and used to explicitly calculate some homology groups and some problems about universal enveloping algebras and actions will be solved. On the other hand, we will focus on giving categorical characterisations of some algebraic structures, such as a characterisation of groups amongst monoids, of cocommutative Hopf algebras amongst cocommutative bialgebras and of Lie algebras amongst alternating algebras.
Full text
Categorical-algebraic methods in non-commutative and non-associative algebra XABIER GARCÍA MARTÍNEZ 2017
Categorical-algebraic methods in non-commutative and non-associative algebra by XABIER GARCÍA MARTÍNEZ DOCTORAL DISSERTATION Submitted for the degree of DOCTOR EN MATEMÁTICAS en la UNIVERSIDAD DE SANTIAGO DE COMPOSTELA Santiago de Compostela, 2017
Categorical-algebraic methods in non-commutative and non-associative algebra Fdo.: Xabier García Martínez Memoria para optar al grado de Doctor realizada en el Departamento de Matemáticas en el Programa de Doctorado en Matemáticas, de la Facultad de Matemáticas de la Universidad de Santiago de Compostela, bajo la dirección de los Profesores Dr. Manuel Ladra González y Dr. Tim Van der Linden Santiago de Compostela, a 01 de septiembre de 2017. Fdo.: Manuel Ladra González Fdo.: Tim Van der Linden
Categorical-algebraic methods in non-commutative and non-associative algebra AUTORIZACIÓN DEL DIRECTOR/TUTOR DE LA TESIS Dr. Manuel Ladra González, Profesor del Departamento de Matemáticas de la Universidad de Santiago de Compostela y Dr. Tim Van der Linden, Profesor en el Institut de Recherche en Mathématique et Physique de la Université Catholique de Louvain (Bélgica), como Directores de la Tesis Doctoral titulada Categorical-algebraic methods in non-commutative and nonassociative algebra, presentada por D. Xabier García Martínez, alumno del programa de Doctorado en Matemáticas, AUTORIZAMOS la presentación de la Tesis Doctoral indicada para optar al grado de Doctor por la Universidad de Santiago de Compostela, con mención internacional, considerando que reúne los requisitos exigidos en el artículo 34 del reglamento de Estudos de Doutoramento, y que como Directores de la misma no incurre en causas de abstención establecidas en la ley 40/2015. Santiago de Compostela, a 01 de septiembre de 2017. Fdo.: Manuel Ladra González Fdo.: Tim Van der Linden
Os resultados presentados nesta memoria foron obtidos grazas a unha bolsa FPU do Ministerio de Educación, Cultura y Deporte con referencia FPU13/01248, a unha bolsa de axuda á mobilidade da Fundación Barrié, aos proxectos MTM2013-43687-P e MTM2016-79661-P do Ministerio de Economía y Competitividad e Agencia Estatal de Investigación (incluído cofinanciamento do FEDER) e ao grupo GRC2013-045 Consellería de Cultura, Educación e Ordenación Universitaria da Xunta de Galicia, na modalidade de Grupo de Referencia Competitiva (incluído cofinanciamento do FEDER) Unión Europea Fondo Europeo de Desarrollo Regional FPU13/01248 Ministerio de Educación, Cultura y Deporte Fundación Barrié MTM2013-43687-P Ministerio de Economía y Competitividad MTM2016-79661-P Agencia Estatal de Investigación GRC2013-045 Xunta de Galicia
Introduction Ever since the introduction of abelian categories in the 1950s as a categorical abstraction of the properties of abelian groups and modules, there has been the wish to find a similar framework that nicely reflects the properties of (not necessarily abelian) groups, rings and algebras. Over the years, several attempts at such an abstract framework have been made: worth mentioning here is the work of Higgins [24], Huq [26] and R.- Grandjeán [32], see [27] for an exhaustive list. Since none of these attempts were entirely successful, and furthermore the connections between them were not clear, some of these approaches were not further developed or even given a name. In 1999, Janelidze, Márki and Tholen realised that Barr-exactness [2], combined with the concept of Bourn-protomodularity [5], provides a context which simplifies and unifies the above-mentioned “old” axiom systems, and in which the relationships with modern categorical algebra can be explored. Expressed in terms of “new” axioms, a semi-abelian category is a pointed category which is Barr-exact and Bourn-protomodular with finite sums. Examples of semi-abelian categories are abundant and ubiquitous. In particular, we may find many of the non-associative and non-commutative algebraic structures studied in the literature [3], including all those that have an underlying group structure. More precisely, any pointed variety of algebras which has amongst its operations and identities those of the theory of groups is semiabelian. One of the advantages of this categorical framework is that it allows a unified study of many important homological properties. For instance, in any semi-abelian category, the classical diagram lemmas (the Short Five Lemma, the 3ˆ3Lemma, the Snake Lemma,Noether’s Isomorphism Theorems) hold. As seen in [34], the theory of semi-abelian categories is perfectly suited for the study of non-abelian (co)homology and the corresponding homotopy theory, xvii
unifying many basic aspects of the classical (co)homology theories of groups, Lie algebras and crossed modules. From an algebraic point of view, the cohomology theory of Lie algebras was introduced in [11], aiming to give an algebraic construction of the cohomology of topological spaces of compact Lie groups. It was very much studied through the years and extended to other related structures, such as crossed modules of Lie algebras [8, 7], Lie superalgebras [31], Lie-Rinehart algebras [25, 33], Leibniz (super)algebras [28], n-Lie algebras [1], n-Leibniz algebras [9], etc. The theory of non-associative algebras is strongly related to different areas of mathematics and it has many applications in physics, mechanics, biology and other sciences. Foremost amongst them are the theories of Lie and Jordan algebras, which have had an enormous relevance in the past century. The study of non-associative algebras encompasses the theory of not necessarily associative R-algebras (with associative algebras being an important special case), where Rmay be a ring or a field. The problems arising in these topics are of various kinds, such as the study of solvability and nilpotency, classifications, characterisations, relations with differential geometry and manifolds, etc. The objective of this dissertation is twofold: firstly to use categorical and algebraic methods to study homological properties of some of the aforementioned semi-abelian, non-associative structures and secondly to use categorical and algebraic methods to study categorical properties and provide categorical characterisations of some well-known algebraic structures. On one hand, the theory of universal central extensions together with the non-abelian tensor product will be studied and used to explicitly calculate some homology groups [10, 12, 17, 19, 18] and some problems about universal enveloping algebras and actions will be solved [14, 15, 6, 20]. On the other hand, we will focus on giving categorical characterisations of some algebraic structures, such as a characterisation of groups amongst monoids [16], of cocommutative Hopf algebras amongst cocommutative bialgebras [22] and of Lie algebras amongst alternating algebras [21]. Since each chapter will have an explicit and fully detailed introduction it does not seem necessary to overextend this first general overview. It is worth mentioning here that notations might not be coherent throughout the text, nevertheless the notation in each chapter is fully internally consistent. The dissertation is organised as follows: In Chapter 1, the universal central extension of a Lie-Rinehart algebra is described and related with the generalisation of Ellis’s non-abelian tensor product [13] of Lie algebras to Lie-Rinehart xviii
algebras. Chapter 2 is devoted to introducing the non-abelian tensor product of Lie superalgebras, relating it with universal central extensions. Here also the low-dimensional non-abelian homology is introduced and its relationship with the cyclic homology of associative superalgebras is established. In Chapter 3 an explicit computation of H2`slpm, n, Aq˘and H2`stpm, n, Aq˘is obtained, where Ais a superalgebra and 3ďm`nď5, and connections with the cyclic homology of associative superalgebras are made. Later on, these results are extended in Chapter 4 to the case of superdialgebras for any m`ně3, with the additional interest of introducing a new method using the non-abelian tensor product. A generalisation of Ellis’s non-abelian exterior product [13] to Leibniz algebras is given in Chapter 5, where it is applied to the construction of an eight term exact sequence in Leibniz homology. Chapter 6 is devoted to extending the notion of biderivation of Leibniz algebras to the crossed modules setting, and to check in which situations it behaves as the actor of the category (also called the split extension classifier in [4]). In Chapter 7, the universal enveloping algebra of a crossed module of Leibniz algebras is studied using new techniques. Then, it is seen as a particular case of crossed modules of Lie algebras in the Loday-Pirashvili category [29]. In Chapter 8 a proper definition of the universal enveloping algebra functor of n-Lie algebras is given, and it is proven that this functor cannot have a right adjoint. In Chapter 9 there is a sharpened version of the characterisation of groups amongst monoids given by Montoli, Rodelo and Van der Linden in [30], proving that a monoid is a group if and only if all splits extensions over it are strong. This characterisation is generalised in Chapter 10, where the following result is obtained: a cocommutative bialgebra over an algebraically closed field is a Hopf algebra if and only if all splits extensions over it are stably strong. In this chapter it is also shown that the category of (not necessarily commutative or cocommutative) Hopf algebras is not unital, so in particular is not semi-abelian. Using Gray’s notion of locally algebraically cartesian closed category [23], in Chapter 11 a characterisation of Lie algebras amongst all varieties of non-associative, alternating algebras is given. The result is a categorical characterisation of the Jacobi identity. Finally, Chapter 12 is devoted to saying some words about the current state of affairs of several works in progress that are taking shape right now, and to mention some lines of research that may be followed when taking this dissertation as a starting point. xix
Bibliography [1] J. Arnlind, A. Kitouni, A. Makhlouf, and S. Silvestrov, Structure and cohomology of 3-Lie algebras induced by Lie algebras, Geometry and Mathematical Physics (AGMP, Mulhouse, France), Springer Proceedings in Mathematics & Statistics, vol. 85, 2014, pp. 123–144. [2] M. Barr, Exact categories, Exact Categories and Categories of Sheaves, Lecture Notes in Math., vol. 236, Springer, Berlin, 1971, pp. 1–120. [3] F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semiabelian categories, Math. Appl., vol. 566, Kluwer Acad. Publ., 2004. [4] F. Borceux, G. Janelidze, and G. M. Kelly, Internal object actions, Comment. Math. Univ. Carolin. 46 (2005), no. 2, 235–255. [5] D. Bourn, Normalization equivalence, kernel equivalence and affine categories, Category theory (Como, 1990), Lecture Notes in Math., vol. 1488, Springer, Berlin, 1991, pp. 43–62. [6] J. M. Casas, R. Fernández-Casado, X. García-Martínez, and E. Khmaladze, Actor of a crossed module of Leibniz algebras, preprint arXiv:1606.04871, 2016. [7] J. M. Casas, N. Inassaridze, and M. Ladra, Homological aspects of Lie algebra crossed modules, Manuscripta Math. 131 (2010), no. 3-4, 385– 401. [8] J. M. Casas and M. Ladra, Homology of crossed modules in Lie algebras, Bull. Soc. Math. Belg. Sér. A 45 (1993), no. 1-2, 59–84, Third Week on Algebra and Algebraic Geometry (SAGA III) (Puerto de la Cruz, 1992). [9] J. M. Casas, J.-L. Loday, and T. Pirashvili, Leibniz n-algebras, Forum Math. 14 (2002), no. 2, 189–207. [10] J. L. Castiglioni, X. García-Martínez, and M. Ladra, Universal central extensions of Lie-Rinehart algebras, J. Algebra Appl., 2017, doi:10.1142/S0219498818501347. [11] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. xx
[12] G. Donadze, X. García-Martínez, and E. Khmaladze, A non-abelian exterior product and homology of Leibniz algebras, Rev. Mat. Complut., 2017, doi:10.1007/s13163-017-0237-2. [13] G. J. Ellis, A nonabelian tensor product of Lie algebras, Glasgow Math. J. 33 (1991), no. 1, 101–120. [14] R. Fernández-Casado, X. García-Martínez, and M. Ladra, A natural extension of the universal enveloping algebra functor to crossed modules of Leibniz algebras, Appl. Categ. Structures, 2016, doi:10.1007/s10485-0169472-9. [15] R. Fernández-Casado, X. García-Martínez, and M. Ladra, Universal enveloping crossed module of Leibniz crossed modules and representations, Journal of Physics: Conference Series 697 (2016), no. 1, 012007. [16] X. García-Martínez, A new characterisation of groups amongst monoids, Appl. Categ. Structures 25 (2017), no. 4, 659–661. [17] X. García-Martínez, E. Khmaladze, and M. Ladra, Non-abelian tensor product and homology of Lie superalgebras, J. Algebra 440 (2015), 464– 488. [18] X. García-Martínez and M. Ladra, Universal central extensions of Leibniz superalgebras over superdialgebras, Mediterr. J. Math. 14 (2017), no. 2, Art. 73, 15. [19] X. García-Martínez and M. Ladra, Universal central extensions of slpm, n, Aqof small rank over associative superalgebras, Turkish J. Math., 2017, doi:10.3906/mat-1604-3. [20] X. García-Martínez, R. Turdibaev, and T. Van der Linden, Do n-Lie algebras have universal enveloping algebras, J. Lie Theory 28 (2018), no. 1, 43–55. [21] X. García-Martínez and T. Van der Linden, A characterisation of Lie algebras amongst alternating algebras, preprint arXiv:1701.05493, 2017. [22] X. García-Martínez and T. Van der Linden, A note on split extensions of bialgebras, preprint arXiv:1701.00665, 2017. xxi
[23] J. R. A. Gray, Algebraic exponentiation in general categories, Ph.D. thesis, University of Cape Town, 2010. [24] P. J. Higgins, Groups with multiple operators, Proc. London Math. Soc. (3) 6(1956), 366–416. [25] J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57–113. [26] S. A. Huq, Commutator, nilpotency, and solvability in categories, Quart. J. Math. Oxford Ser. (2) 19 (1968), 363–389. [27] G. Janelidze, L. Márki, and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002), no. 2-3, 367–386, Category theory 1999 (Coimbra). [28] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), no. 1, 139–158. [29] J.-L. Loday and T. Pirashvili, The tensor category of linear maps and Leibniz algebras, Georgian Math. J. 5(1998), no. 3, 263–276. [30] A. Montoli, D. Rodelo, and T. Van der Linden, Two characterisations of groups amongst monoids, J. Pure Appl. Algebra, 2016, doi:10.1016/j.jpaa.2017.05.005. [31] I. M. Musson, Lie superalgebras and enveloping algebras, Graduate Studies in Mathematics, vol. 131, American Mathematical Society, Providence, RI, 2012. [32] A. R.-Grandjeán, Homología en categorías exactas, Ph.D. thesis, Universidade de Santiago de Compostela, 1970. [33] G. S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195–222. [34] T. Van der Linden, Homology and homotopy in semi-abelian categories, Ph.D. thesis, Vrije Universiteit Brussel, 2006. xxii
Chapter 1 Universal central extensions of Lie-Rinehart algebras Abstract In this paper we study the universal central extension of a Lie-Rinehart algebra and we give a description of it. Then we study the lifting of automorphisms and derivations to central extensions. We also give a definition of a non-abelian tensor product in Lie-Rinehart algebras based on the construction of Ellis of non-abelian tensor product of Lie algebras. We relate this non-abelian tensor product to the universal central extension. Reference J. L. Castiglioni, X. García-Martínez, and M. Ladra, Universal central extensions of Lie-Rinehart algebras, J. Algebra Appl., 2017, doi:10.1142/S0219498818501347. 1.1 Introduction Let A be a unital commutative algebra over a commutative ring Kwith unit. A Lie-Rinehart algebra is a Lie K-algebra, which is also an A-module and these two structures are related in an appropriate way [11]. The leading example of Lie-Rinehart algebras is the set DerKpAqof all K-derivations of A. LieRinehart algebras are the algebraic counterpart of Lie algebroids [20]. 1
2 1 Universal central extensions of Lie-Rinehart algebras The concept of Lie-Rinehart algebra generalizes the notion of Lie algebra. In [24] and [8] universal central extensions of Lie algebra are studied, proving that if a Lie algebra is perfect then it has a universal central extension. Moreover, it is characterized the kernel of the universal central extension as the second homology group with trivial coefficients. In this paper we extend this study to Lie-Rinehart algebras. On the other hand, in [6] a non-abelian tensor product of Lie algebras is introduced, its more important properties are studied and it is related to the universal central extension. It has been extended to other structures as Leibniz algebras [9], Lie superalgebras [7] or Hom-Lie algebras [3]. In this paper we broaden this construction to Lie-Rinehart algebras, we study some important properties and we relate it to the universal central extension of Lie-Rinehart algebras. After the introduction, the paper is organized in four sections. In Sec. 1.2, we recall some needed notions and facts on Lie-Rinehart algebras, actions, crossed modules, universal enveloping algebras, free algebras, homology and cohomology and abelian extensions of Lie-Rinehart algebras. In Sec. 1.3, following Neher’s paper on Lie superalgebras [21], we introduce central extensions and universal central extensions of Lie-Rinehart algebras giving a characterization of them (Theorem 1.3.7), extending classic results of Lie algebras (see [24]). We construct an endofunctor uceAthat when the Lie-Rinehart algebra is perfect gives explicitly the universal central extension. In Sec. 1.4, we study the lifting of automorphisms and derivations to central extensions. Finally, in Sec. 1.5, we introduce a non-abelian tensor product of Lie-Rinehart algebras extending Ellis [6] non-abelian tensor product of Lie algebras. We relate this non-abelian tensor product with the universal central extension. 1.2 Preliminaries on Lie-Rinehart algebras Most of the content of this section is well known, or follows from known results (see [4, 5, 11, 23]). We included it in order to fix terminology, notations and main examples. In what follows we fix a unital commutative ring K. All modules are considered over K. We write band Hom instead of bKand HomK.
1.2 Preliminaries on Lie-Rinehart algebras 3 1.2.1 Definitions, Examples Let A be a unital commutative algebra over K. Then the set DerKpAqof all K-derivations of A is a Lie K-algebra and an A-module simultaneously. These two structures are related by the following identity rD, aD1s “ arD, D1s ` DpaqD1, D, D1PDerKpAq. This leads to the notion below, which goes back to Herz under the name “pseudo-algèbre de Lie” and which is the algebraic counterpart of the Lie algebroid [20]. Definition 1.2.1. ALie-Rinehart A-algebra consists of a Lie K-algebra L together with an A-module structure on Land a morphism, called the anchor map, α:LÑDerKpAq, which is simultaneously a Lie algebra and A-module homomorphism such that rx, ays “ arx, ys ` xpaqy. Here x, y PL,aPAand we write xpaqfor αpxqpaq[11]. These objects are also known as pK, Aq-Lie algebras [23] and d-Lie rings [22]. As stated in the literature ([10] for example), if Lis a faithful A-module, the requirement of α being a Lie homomorphism follows from the other axioms. Thus DerKpAqwith α“IdDerKpAqis a Lie-Rinehart A-algebra. Let us observe that Lie-Rinehart A-algebras with trivial homomorphism α:LÑ DerKpAqare exactly Lie A-algebras. Therefore the concept of Lie-Rinehart algebras generalizes the concept of Lie A-algebras. If A“K, then DerKpAq=0 and there is no difference between Lie and Lie-Rinehart algebras. If Lis an A-module, then Lis a trivial Lie-Rinehart A-algebra, that is Litself endowed with trivial Lie bracket and trivial anchor map. If Land L1are Lie-Rinehart algebras, a Lie-Rinehart A-algebra homomorphism f:LÑL1is a map, which is simultaneously a Lie K-algebra homomorphism and a homomorphism of A-modules. Furthermore it has to preserve the action on DerKpAq, in other words the diagram Lf,2 α( L1 α1 v DerKpAq
10 1 Universal central extensions of Lie-Rinehart algebras such that CnCmĂCn`mand such that the associated graded object gr˚pCq “ Àně0Cn{Cn´1is a commutative A-algebra. Remark 1.2.13.If Cis an almost commutative algebra, then there is a welldefined bracket r´,´s:grnpCq b grmpCq ÝÑ grn`m´1pCq which is given as follows. Let aPgrnpCqand bPgrmpCqand ˆaPCnand ˆ bPCmbe representatives of aand b. Since gr˚pCqis a commutative algebra it follows that ˆaˆ b´ˆ bˆaPCn`m´1and the corresponding class in grn`m´1pCq is ra, bs. In this way we obtain a Poisson algebra structure on gr˚pCq. Since the bracket is of degree (-1) it follows from Example 1.2.8 that L“gr1pCqis a Lie-Rinehart A“gr0pCq-algebra. Moreover the short exact sequence AÑC1ÑL is an abelian extension of Lie-Rinehart algebras (see below Definition 1.2.18). Proposition 1.2.14. The correspondence assigning C1to the almost commutative algebra C, defines a functor LR:ACommAÑLRAK. Proof. Let f:CÑDbe a morphism in ACommA. Since fpreserves the filtration, fpC1q Ď D1. Furthermore, fpaxq “ fpaqfpxq “ afpxq, for any aPC0“D0and xPC1, and fprx, ysq “ fpxy ´yxq “ fpxqfpyq ´ fpyqfpxq “ rfpxq, fpyqs, for x, y PC1. Hence the restriction of fto C1, which we shall call LRpfq, is a morphism of K-Lie algebras and of A-modules such that the following diagram commutes in LieK, C1 LRpfq,2 r˝,´s ) D1 r˝,´s u DerKpAq. Thus, LRpfq P LRAK. On the other hand, it is clear that LRp1C1q “ 1C1and the following diagram commutes in K-mod, Cf,2Dg,2E C1 ? iC LR LRpfq,2D1 ? iD LR LRpgq ,2E1. ? iE LR
1.2 Preliminaries on Lie-Rinehart algebras 11 Hence LR is functorial. Proposition 1.2.15. The functor LR is right adjoint to the universal enveloping functor UA:LRAKÑACommA. Proof. Let Φ: ACommApUAL,Cq Ñ LRAK`L,LRpCq˘be the map given as follows. Since UALis generated as a K-algebra by Land A, a morphism f:UALÑCis completely determined by its restriction to Land A. Since fpaq “ afor every aPA, and fpLq Ď f`pUALq1˘ĎC1, it follows that the restriction of fto L,Φf:LÑC1“LRpCqis a morphism of Lie-Rinehart algebras and Φis a monomorphism. Let g:LÑC1be a morphism in LRAK. We build up rg:UALÑCby r gpax1¨ ¨ ¨ xmq:“agpx1q ¨ ¨ ¨ gpxmq P C. It is straightforward to see that rgis a morphism in ACommAand Φr g“g. Hence Φis bijective, and UAand LR form an adjoint pair. There is another way to understand the universal enveloping algebra as an adjunction. We consider the category AncAof anchored algebras, defined as A-algebras Bequipped with an A-algebra morphism α:BÑEndpAq, where the A-algebra structure on EndpAqis given by aÞÑ pla:a1ÞÑ aa1qand we construct a functor from AncAto LRAKthat sends an anchored algebra Bto the A-submodule consisting of those elements bPBsuch that αpbq P DerKpAq. Then this functor is left adjoint to the universal enveloping functor (see [1]). 1.2.5 Free Lie-Rinehart Algebras Here we follow [5]. Let Kmod{DerKpAqbe the category of K-linear maps ψ:VÑDerKpAq, where Vis a K-module. We have the functor U:LRAKÑKmod{DerKpAq which assigns α:LÑDerKpAqto a Lie-Rinehart algebra L. A morphism ψÑψ1in Kmod{DerKpAqis a K-linear map f:VÑV1such that ψ“ψ1f. Now we construct the functor F:Kmod{DerKpAq Ñ LRAK as follows. Let ψ:VÑDerKpAqbe a K-linear map. We let LpVqbe the free Lie K-algebra generated by V. Then we have the unique Lie K-algebra
12 1 Universal central extensions of Lie-Rinehart algebras homomorphism LpVq Ñ DerKpAqwhich extends the map ψ, which is still denoted by ψ. Now we can apply the construction from Example 1.2.3 to get a Lie-Rinehart algebra structure on AbLpVq. We let Fpψqbe this particular Lie-Rinehart algebra and we call it the free Lie-Rinehart algebra generated by ψ. In this way we obtain the functor F, which is the left adjoint to U. Kapranov [17] defines a different concept of free Lie-Rinehart algebra as the adjoint of the forgetful functor U1:LRAKÑAmod{DerKpAq. The relation between both constructions is given in [17, (2.2.8) Proposition]. 1.2.6 Rinehart homology and cohomology of Lie-Rinehart algebras Let Mbe a left Lie-Rinehart pA, Lq-module. Let us recall the definition of the Rinehart cohomology H˚ RinpL, Mqof a Lie-Rinehart algebra Lwith coefficients in a left Lie-Rinehart module M(see [22, 23] and [5, 11]). We put Cn ApL, Mq:“HomApΛn AL, Mq, n ě0, where Λ˚ ApVqdenotes the exterior algebra over A generated by an A-module V. The coboundary map δ:Cn´1 ApL, Mq ÝÑ Cn ApL, Mq, is given by pδfqpx1, . . . , xnq “ n ÿ i“1 p´1qpi´1qxi`fpx1, . . . , ˆxi, . . . , xnq˘ `ÿ jăk p´1qj`kfprxj, xks, x1, . . . , ˆxj, . . . , ˆxk, . . . , xnq, where x1, . . . , xnPL, m PM, f PCn´1 ApL, Mq. We note that the differential δis not A-linear unless Lacts trivially on A. For any left Lie-Rinehart pA, Lq-module M, the Lie-Rinehart cohomology is defined by Hn RinpL, Mq “ Hn`Cn ApL, Mq˘, n ě0. Let Mbe a right Lie-Rinehart pA, Lq-module. Let us recall the definition of the Rinehart homology HRin ˚pL, Mqof a Lie-Rinehart algebra Lwith coefficients in a right Lie-Rinehart module M. We put CA npL, Mq:“MbAΛn AL, n ě0.
1.2 Preliminaries on Lie-Rinehart algebras 13 The boundary map B:CA npL, Mq ÝÑ CA n´1pL, Mq, is given by B`mbApx1, . . . , xnq˘“ n ÿ i“1 p´1qpi´1qmxibApx1, . . . , ˆxi, . . . , xnq `ÿ jăk p´1qj`kmbAprxj, xks, x1, . . . , ˆxj, . . . , ˆxk, . . . , xnq, where x1, . . . , xnPL, m PM. We note that the differential Bis not A-linear unless Lacts trivially on A. For any right Lie-Rinehart pA, Lq-module M, the Lie-Rinehart homology is defined by HRin npL, Mq “ Hn`CA npL, Mq˘, n ě0. Let gbe a Lie algebra over Kand let Mbe a g-module. Then we have the Chevalley-Eilenberg chain and cochain complexes CLie ˚pg,Mqand C˚ Liepg,Mq, which compute the Lie algebra (co)homology (see [2]): CLie npg,Mq “ Λnpgq b M, Cn Liepg,Mq “ HompΛnpgq,Mq. Here Λ˚denotes the exterior algebra defined over K. One observes that if A“K, then HRin ˚pL, Mqand H˚ RinpL, Mqgeneralize the classical definition of Lie algebra (co)homology. For a general A by forgetting the A-module structure one obtains the canonical homomorphisms HLie ˚pL, Mq Ñ HRin ˚pL, Mq, H˚ RinpL, Mq Ñ H˚ LiepL, Mq, where HLie ˚pL, Mqand H˚ LiepL, Mqdenote the homology and cohomology of L considered as a Lie K-algebra. On the other hand if A is a smooth commutative algebra, then H˚ RinpDerpAq,Aqis isomorphic to the de Rham cohomology of A (see [23] and [11]). Lemma 1.2.16. Let gbe a Lie K-algebra acting on a commutative algebra A by derivations and let Lbe the transformation Lie-Rinehart algebra of pg,Aq
14 1 Universal central extensions of Lie-Rinehart algebras (see Example 1.2.3). Then for any (right or left, depending on the situation) Lie-Rinehart pA, Lq-module Mwe have the canonical isomorphisms of complexes CA ˚pL, Mq – CLie ˚pg,Mq,C˚ ApL, Mq – Cn Liepg,Mqand in particular the isomorphisms HRin ˚pL, Mq – HLie ˚pg,Mq, H˚ RinpL, Mq – H˚ Liepg,Mq. Proof. Since L“Abgwe have Λn ALbAM–ΛngbMand HomApΛn AL, Mq – HompΛng,Mqand lemma follows. Proposition 1.2.17. Let Lbe a free Lie-Rinehart algebra generated by ψ:VÑDerKpAqand let Mbe any (right or left, depending on the situation) Lie-Rinehart pA, Lq-module. Then HRin npL, Mq “ 0, n ą1, Hn RinpL, Mq “ 0, n ą1. Proof. By our construction Lis a transformation Lie-Rinehart algebra of pLpVq,Aq. Thus we can apply Lemma 1.2.16 to get isomorphisms HRin ˚pL, Mq – HLie ˚pLpVq,Mqand H˚ RinpL, Mq – H˚ LiepLpVq,Mqand then we can use the well-known vanishing result for free Lie algebras (see [25]). 1.2.7 Low degree homology groups of Lie-Rinehart algebras Here we follow [5]. By definition, HRin 0pL, Mq “ M M˝L, is the module of coinvariants of M, where M˝Lmeans the K-submodule of Mgenerated by mx,xPL, m PM, and H0 RinpL, Mq “ ML“ tmPM|xm “0for all xPLu, is the invariant K-submodule of M. It follows from the definition that one has the following exact sequence 0ÑH0 RinpL, Mq Ñ Md ÝÝÑ DerApL, Mq Ñ H1 RinpL, Mq Ñ 0,(1.2.1) where DerApL, Mqconsists of A-linear maps d:LÑMwhich are derivations from the Lie K-algebra Lto M. In other words dmust satisfy the following conditions: dpaxq “ adpxq, dprx, ysq “ x`dpyq˘´y`dpxq˘, a PA, x, y PL.
1.3 Universal central extensions of Lie-Rinehart algebras 15 If mPM, the map dm:LÑM, x ÞÑ xm, is a derivation. The maps dmare called inner derivations of Linto M, and they form an K-submodule IDerApL, Mqof DerApL, Mq. By (1.2.1), H1 RinpL, Mq – DerApL, Mq{ IDerApL, Mq. If Mis a trivial pA, Lq-module, then H1 RinpL, Mq – DerApL, Mq – HomApLab,Mqand HRin 1pL, Mq – MbAL MbAtL,Lu–MbALab. 1.2.8 Abelian extensions of Lie-Rinehart algebras Definition 1.2.18. Let Lbe a Lie-Rinehart A-algebra and let Ma left LieRinehart pA, Lq-module. An abelian extension of Lby Mis a short exact sequence Mi ÝÑ L1B ÝÝÑ L, where L1is a Lie-Rinehart A-algebra and Bis a Lie-Rinehart algebra homomorphism. Moreover, iis an A-linear map and the following identities hold ripmq, ipnqs “ 0, ripmq, x1s “ `Bpx1q˘pmq, m, n PM, x1PL1. An abelian extension is called A-split if Bhas an A-linear section. Proposition 1.2.19 ([11, Theorem 2.6]).If Lis A-projective, then the cohomology H2 RinpL, Mqclassifies the abelian extensions MÝÑ L1ÝÑ L of Lby Min the category of Lie-Rinehart algebras that split in the category of A-modules. The extension MÝÑ L‘MÝÑ Lrepresents 0PH2 RinpL, Mq. 1.3 Universal central extensions of Lie-Rinehart algebras 1.3.1 Central extensions An extension of a Lie-Rinehart algebra Lis a short exact sequence Ii ÝÑ Ep ÝÝÑ L, (1.3.1)
16 1 Universal central extensions of Lie-Rinehart algebras where I, E and Lare Lie-Rinehart algebras and i, p are Lie-Rinehart homomorphisms. Since i:IÑipIq “ Ker pis an isomorphism we shall identify I and ipIq. In other words, an extension of Lis a surjective Lie-Rinehart homomorphism p:EÑL. If p:EÑLand p1:E1ÑLare two extensions of L, a homomorphism from pto p1is a commutative diagram in LRAKof the form Ef,2 p% % E1 p1 xx L . In particular, Ker fĎf´1pKer p1q “ Ker pand E1“fpEq ` Ker p1.(1.3.2) An extension (1.3.1) is called split if there exists a Lie-Rinehart morphism s:LÑE, called splitting homomorphism, such that ps “1L. In this case, E“I‘spLqand s:LÑspLqis an isomorphism with inverse p|spLq. Moreover, E»I¸L, the semidirect product. In this way, semidirect products and split exact sequences are in a one to one correspondence. We point out that not every extension splits. We shall say that an extension splits uniquely whenever the splitting morphism is unique. Acentral extension of Lis an extension psuch that Ker pĎZApEq. In particular, if p:Ep,2,2L s qxis a split central extension, it is a product of K-Lie algebras E“Ker pˆL, which is also a Lie-Rinehart algebra. Proposition 1.3.1. If Lis A-projective, then H2 RinpL, Iqclassifies the central extensions IÝÑ EÝÑ L of Lby I. Proof. Note that, if Iis a trivial left Lie-Rinehart pA, Lq-module, then an abelian extension of Lby Iis a central extension, and so the assertion follows by Proposition 1.2.19. In the A-projective case, the study of central extensions of Lie-Rinehart algebras can be seen as central extensions of Lie algebras, as we show in the next proposition.
1.3 Universal central extensions of Lie-Rinehart algebras 17 Proposition 1.3.2. Let I“A¨αpLqpAqbe the ideal generated by the action of the anchor map on A. Then L“L{IL is a Lie algebra over A“A{I, and there is an equivalence of categories between on the one hand central extensions of the Lie-Rinehart algebra Lthat split as A-modules, and on the other hand central extensions of the A-Lie algebra L“L{IL. Proof. If Ker p,2Ep,2,2Lis an A-split central extension, then centrality of Ker pimplies I¨Ker p“ t0u. Now I¨LĂLis an ideal of K-Lie algebras, making L“L{IL a Lie-Rinehart algebra over Awith trivial anchor map, that is, an A-Lie algebra. Since pis A-split, we obtain a central extension Ker p,2Ep,2,2Lof A-Lie algebras. Conversely, every central extension Ker p,2˜ Ep,2,2Lof A-Lie algebras gives rise to the pullback central extension ˆ E,2,2 L ˜ Ep,2,2L , where ˆ E“ tpx, yq P ˜ EˆL|xmod Ker p“ymod ILu. The correspondences EÞÑ Eand ˜ EÞÑ ˆ Eare functorial, and the natural transformations implementing the equivalence are the obvious ones. In the rest of this section we will study the general case of central extensions of Lie-Rinehart algebras. A Lie-Rinehart A-algebra Lis said perfect if L“ tL, Lu. A central extension Eof Lis called a covering if Eis perfect; in that case, Lis also perfect. A central extension u:LÑLis called universal if there exists a unique homomorphism from uto any other central extension of L. From the universal property of universal central extensions it immediately follows that two universal central extensions of Lare isomorphic as extensions. Lemma 1.3.3. (central trick) Let p:EÝÝLbe a central extension. (a) If ppxq “ ppx1qand ppyq “ ppy1qthen rx, ys “ rx1, y1sand for every aPA, xpaq “ x1paq.
18 1 Universal central extensions of Lie-Rinehart algebras (b) If the following diagram commutes in LRAK, P f,2 g,2Ep,2,2L then the restriction of both fand gto tP, P uagree; i.e. f|tP,P u“g|tP,P u. In particular, there exists at most one homomorphism from a covering CÝÝLto the central extension Ep ÝÝÝL. Proof. (a) We have x1“x`zand y1“y`z1for some z, z1PKer pĂZApEq, so it is clear that rx1, y1s“rx`z, y `z1s“rx, ys. In addition, if pis a Lie-Rinehart homomorphism, the action on DerKpAqmust be preserved so xpaq “ x1paq. (b) Using part (a), we have gparx, ysq “ argpxq, gpyqs “ arfpxq, fpyqs “ fparx, ysq. Lemma 1.3.4. Let p:EÝÝLbe a central extension where Lis perfect. Then (a) E“ tE, Eu ` Ker p, and p1“p|tE,Eu:tE, Eu ÝÝLis a covering. (b) ZApEq “ p´1`ZApLq˘and p`ZApEq˘“ZApLq. (c) If f:LÝÝMis a central extension then so is fp:EÝÝM. (d) If f:CÝÝLis a covering and Eg,2 p% % C f zz L a morphism of extensions, then g:EÝÝCis a central extension. In particular, gis surjective. Proof. (a) Since pptE, Euq “ tL, Lu “ Lit follows easily that E“ tE, Eu ` Ker pand p|tE,Euis clearly a covering. (b) Let zPZApEq. For every aPAwe have raz, Es “ 0, so 0“ rappzq, ppEqs “ rappzq, Lsthen ppzq P ZApLq. Conversely, let zPp´1`ZApLq˘. For every aPAwe have ppraz, Esq “ rappzq, Ls “ 0so raz, Es Ă Ker pĂ
1.3 Universal central extensions of Lie-Rinehart algebras 19 ZApEq. Since raz, Es “ raz, tE, Eu ` Ker ps “ raz, tE, Eus we just have to check that raz, tE, Eus is zero. Therefore, “az, brx, ys‰“b“az, rx, ys‰“ b“x, raz, ys‰`b“y, rx, azs‰“0. (c) The composition fp is clearly surjective and Ker fp “p´1pKer fq Ă p´1`ZApLq˘“ZApEq. (d) Since C“gpEq ` Ker f, see (1.3.2), we have that C“ tC, Cu “ tgpEq, gpEqu “ gptE, Euq so gis surjective. Moreover, it is central since Ker gĂKer p. Corollary 1.3.5. Let LPLRAK, arbitrary. If L{ZApLqis perfect, then ZApL{ZALq “ 0. Proof. It can be seen applying the second formula of Lemma 1.3.4(b) to the canonical map p:L,2,2L{ZApLq, which is a central extension. Lemma 1.3.6. (pullback Lemma) Let c:NÝÝMbe a central extension and f:LÑMa morphism of Lie-Rinehart algebras, then, P:“ tpl, nq P LˆDerKpAqN|fplq “ cpnqu is a Lie-Rinehart algebra and pL:PÑL,pl, nq ÞÑ l, is a central extension. This extension splits if and only if there exists a (unique) Lie-Rinehart morphism h:LÑNsuch that ch “f. PpN,2 pL N c Lf,2 h 9D s DL M Proof. It is clear that Pis a Lie-Rinehart algebra with action pl, nqpaq “ lpaq “ npaq, and pLis a central extension. Moreover, a splitting homomorphism s:LÑPexists (uniquely) if and only if there exists a (unique) Lie-Rinehart homomorphism h:LÑNsuch that splq “ `l, hplq˘for all lPL. Theorem 1.3.7. (characterization of universal central extensions) For a Lie-Rinehart algebra L, there are equivalent: 1. Every central extension L1ÑLsplits uniquely.
26 1 Universal central extensions of Lie-Rinehart algebras In this way, uceAphqpCq “ uceAphq`fpKer u1q˘“ puceAphqfqpKer u1q “ pf uceAph1qqpKer u1q “ fpKer u1q “ C. Suppose now that uceAphqpCq “ C. We obtain the commutative diagram L uceAphq u1f´1 ,2L1 f,2L h Lu1f´1 ,2L1f,2L . (1.4.2) If uceAphqpCq “ C, the kernel of the epimorphism u1f´1˝uceAphqis C, i.e. the kernel of u1f´1. In this way, we obtain an automorphism h1:L1ÑL1such that (1.4.2) commutes. The condition that h1pKer fq “ Ker ffollows immediately by the commutativity of (1.4.1). (b) The map is well defined and injective by part (a) of the theorem. Let gPAutpL1qsuch that gpKer fq “ Ker f. It descends to hPAutpLqsuch that fg “hf. Again by (a), gmust be the lifting of hand since the lifting exists it follows that uceAphqpCq “ C. Corollary 1.4.2. If Lis perfect, the map AutpLq Ñ tgPAut`uceApLq˘|gpKer uq “ Ker uu, f ÞÑ uceApfq, is a group isomorphism. Moreover, if Lis centreless, then AutpLq – Aut`uceApLq˘. Proof. Applying the last theorem to the covering u:uceApLq Ñ L, we have that u1is the identity map, so C“0and the corollary follows immediately. By Lemma 1.3.4(b), if Lis perfect we have that Ker u“ZA`uceApLq˘and since every automorphism leaves the centre invariant it is straightforward that AutpLq – Aut`uceApLq˘. 1.4.2 Lifting of derivations Definition 1.4.3. Let Lbe a Lie-Rinehart algebra over A. A derivation of L is a pair D:“ pδ, δ0q, where δ:LÑLis a derivation of Las a K-Lie algebra,
1.4 Lifting automorphisms and derivations 27 δ0PDerKpAqand the following identities hold: δpaxq “ aδpxq ` δ0paqx, δ0`xpaq˘“x`δ0paq˘`δpxqpaq, with aPAand xPL. Note that the second identity means that the following diagram commutes, L α δ,2L α DerKpAqrδ0,´s ,2DerKpAq. For any xPLwe have an associated derivation pδ, δ0qwhere δpyq “ rx, ys and δ0paq “ xpaq. We shall write DerpLqthe A-module of all derivations of the LieRinehart algebra L. Observe that DerpLq, with Lie bracket rpδ, δ0q,pδ1, δ1 0qs “ prδ, δ1s,rδ0, δ1 0sq and anchor map DerpLq Ñ DerKpAq,pδ, δ0q ÞÑ δ0is a LieRinehart algebra over A. In the particular case of K“A, we recover the notion of Lie derivation. Recall that if pδ, δ0q P DerpLqwe have that δ`ZApLq˘ĂZApLq, since if xPZApLq, raδpxq, zs “ δprax, zsq ´ rax, δpzqs ´ rδ0paqx, zs “ 0. Lemma 1.4.4. Let f:LÑMbe a central extension of Lie-Rinehart algebras. If pδ, δ0qand pδ1, δ0qare derivations of Lsuch that fδ “fδ1then δ|tL,Lu“ δ1|tL,Lu. Proof. Since fis a central extension, arδpxq, ys “ arδ1pxq, ysand arx, δpyqs “ arx, δ1pyqs. Thus, δparx, ysq “ δ0paqrx, ys ` arδpxq, ys ` arx, δpyqs “δ0paqrx, ys ` arδ1pxq, ys ` arx, δ1pyqs “ δ1parx, ysq, for all aPAand x, y PL. Given a derivation D“ pδ, δ0q P DerpLq, one can define uceApDq “ pδu, δ0q, where δuis defined on generators as pa, x, yq ÞÑ pδ0paq, x, yq ` pa, δpxq, yq `
28 1 Universal central extensions of Lie-Rinehart algebras `a, x, δpyq˘. It is a straightforward verification that the map uceApDqis also a derivation of the Lie-Rinehart algebra uceApLqand yields the following commutative diagram uceApLq u δu,2uceApLq u Lδ,2L leaving Ker uinvariant. Moreover, the map uceA:DerpLq Ñ tpγ, γ0q P Der`uceApLq˘|γpKer uq Ă pKer uqu, D ÞÑ uceApDq, is a Lie-Rinehart homomorphism, and its kernel is contained in the subalgebra of those derivations vanishing on tL, Lu. In the following lemma we check how uceAoperates with derivations. Lemma 1.4.5. Let f:LÑMbe a morphism of perfect Lie-Rinehart algebras and let pδL, δ0q P DerpLqand pδM, δ0q P DerpMqbe such that fδL“δMf. Then, we have that uceApfqδu L“δu MuceApfq. Proof. It suffices to check it for an element pa, x, yq P uceApLq. uceApfqδu Lpa, x, yq “ uceApfq`pδ0paq, x, yq`pa, δLpxq, yq`pa, x, δLpyqq˘ “`δ0paq, fpxq, fpyq˘``a, f`δLpxq˘, fpyq˘``a, fpxq, f`δLpyq˘˘ “`δ0paq, fpxq, fpyq˘``a, δM`fpxq˘, fpyq˘``a, fpxq, δM`fpyq˘˘ “δM`a, fpxq, fpyq˘ “δMuceApfqpa, x, yq. We will state now the analogue of Theorem 1.4.1 for derivations. Theorem 1.4.6. Let f:L1ÑLbe a covering of the Lie-Rinehart algebra L and as before, we denote C“uceApfqpKer u1q. (a) A derivation D“ pδ, δ0q P DerpLqlifts to a derivation D1“ pδ1, δ0qof L1 satisfying δf “fδ1if and only if the derivation δupCq Ă C. Moreover, δ1is uniquely determined and leaves Ker finvariant.
1.4 Lifting automorphisms and derivations 29 (b) The map tpδ, δ0q P DerpLq | uceApδuqpCq Ă Cu ÝÝÑ tpη, η0q P DerpL1q | ηupKer fq Ă Ker fu pδ, δ0q ÞÝÝÑ pδ1, δ0q is an isomorphism of Lie-Rinehart algebras. (c) In particular, from the covering u:uceApLq Ñ Lwe obtain that the map uceA:DerpLq Ñ tpγ, γ0q P Der`uceApLq˘|γpKer uq Ă Ker uu is an isomorphism. Moreover, if Lis centreless, we have that DerpLq – Der`uceApLq˘. Proof. (a) If the derivation D1“ pδ1, δ0qexists, by Lemma 1.4.4 it is unique. Using Lemma 1.4.5 we have that δupCq “ uceApδuq`fpKer u1q˘“ puceApδuqfqpKer u1q “ pf˝uceApδ1uqqpKer u1q Ă fpKer u1q “ C. Conversely, if δupCq Ă C, taking the analogue for derivations of diagram (1.4.1), it follows immediately. (b) The map is well defined and injective by part (a) and surjective by Lemma 1.4.4. (c) We have that u1is the identity map, so C“0. In the case that ZApLq “ 0, we have that Ker u“ZA`uceApLq˘, then DerpLq – Der`uceApLq˘ follows immediately, since every derivation leaves the centre invariant. 1.4.3 Universal central extensions of split exact sequences Theorem 1.4.7. Let Lf,2Mg,2,2N s pwbe a split short exact sequence of perfect Lie-Rinehart algebras. We have the following commutative diagram uceApLqϕ,2 uL uceApMqγ,2,2 uM uceApNq σ nt uN Lf,2Mg,2,2N s ou where uceApMqis a semidirect product uceApMq “ ϕ`uceApLq˘¸σ`uceApNq˘,
30 1 Universal central extensions of Lie-Rinehart algebras and Ker uM“ϕpKer uLq ‘ σpKer uNq. We know that M–L¸Nsince the bottom row exact sequence splits. If moreover M“L‘N, i.e. rfpLq, spNqs “ t0u, we have uceApL‘Nq – uceApLq ‘ uceApNq. Proof. In order to simplify the notation, we can interpret fand sas identifications, so we will write lfor fplqand nfor spnq. Given an element of the form pa, ˜n, ˜ lq P uceApMq, with ˜nPNand ˜ lPL, by the perfectness of Land the properties of uceApMq, we have pa, ˜n, ˜ lq “ pa, brn, n1s, crl, l1sq “ pac, brn, n1s,rl, l1sq ` pabrn, n1spcq, l, l1q “ pac, rbrn, n1s, ls, l1q`pac, l, rbrn, n1s, lsq ` pabrn, n1spcq, l, l1q, which means that pA, N, Lq Ă pA, L, Lq, so uceApMq “ pA, L, Lq ` pA, N, Nq. By definition, pA, L, Lq “ ϕ`uceApLq˘and pA, N, Nq “ σ`uceApNq˘. Now since γσ is the identity map, we know that uceApMq – Ker γ¸σ`uceApNq˘. In this way, σ`uceApNq˘–uceApNqand since pA, L, Lq Ă Ker γit follows that Ker γ“ pA, L, Lq “ ϕ`uceApLq˘, so we have that uceApMq “ ϕ`uceApLq˘¸ σ`uceApNq˘. Every element of uceApMqhas the form ϕplq ` σpnqwhere lPuceApLq and nPuceApNq. This means that any element of uceApMqis in Ker uMif and only if 0“uMϕplq “ uLplqand 0“uMσpnq “ uNpnq, so Ker uM“ ϕpKer uLq ‘ σpKer uNq. In the particular case that M“L‘N, we can define the induced map ϕ‘σ:uceApLq ‘ uceApNq Ñ uceApMq, and it is an easy computation that it is a Lie-Rinehart algebra morphism. Moreover, Kerpϕ‘σq “ Ker ϕ. Given lPKer ϕ,uMϕplq “ 0“uLplqso lPKer uLPZApLq, which means that ϕ‘σis a covering. We can use now Theorem 1.3.7(2) to see that ϕ‘σis an isomorphism completing the proof.
1.5 Non-abelian tensor product of Lie-Rinehart algebras 31 1.5 Non-abelian tensor product of Lie-Rinehart algebras A non-abelian tensor product of Lie algebras was introduced by Ellis [6]. Here we adapt some of his results to the case of Lie-Rinehart algebras, in order to use them to obtain a description of universal central extensions in this category. Definition 1.5.1. Let L, M PLRAK. By a quasi-action of Lon M, we mean aK-bilinear map, LˆMÑM,px, mq ÞÑ xm, satisfying 1. xpamq “ apxmq ` xpaqm, 2. rx,ysm“xpymq ´ ypxmq, 3. xrm, ns“rxm, ns ` rm,xns, for all aPA,x, y PLand m, n PM. We will say that Lquasi-acts on M. For example, if Lis a subalgebra of some Lie-Rinehart algebra Land Mis an ideal of Lthen the bracket in Lyields a quasi-action of Lon M. In the particular case of K“A, a quasi-action is the same as a Lie action. Remark 1.5.2.The category of Lie-Rinehart algebras does not fit into the theory of semi-abelian categories in the sense of [15] so the notion of internal action [14] cannot be recovered. We could be tempted to add another identity such as axm“apxmq(identity (4) in Definition 1.2.9), but then a Lie-Rinehart algebra would not act on itself via the bracket. On the other hand, since the normal subobjects are very limited (they have to be Lie A-algebras), to form a semidirect product compatible with the notion of split extensions we recover the notion of action of Definition 1.2.9, but this is not useful to our proposes since we cannot form an action over an arbitrary Lie-Rinehart algebra. If we have a quasi-action of Lon Mand a quasi-action of Mon L, for any Lie-Rinehart algebra Lwe call a K-bilinear function f:LˆMÑLa Lie-Rinehart pairing if 1. αL`fpx, mq˘“ rαLpxq, αMpmqs, 2. fprx, ys, mq “ fpx, ymq ´ fpy, xmq, 3. fpx, rm, nsq “ fpnx, mq ´ fpmx, nq,
32 1 Universal central extensions of Lie-Rinehart algebras 4. f`apmxq, bpynq˘“ ´abrfpx, mq, fpy, nqs ´ arαLpxq, αMpmqspbqfpy, nq ` rαLpyq, αMpnqspaqbfpx, mq, for all a, b PA,x, y PL,m, n PM, and where αL,αMdenote the anchor maps corresponding to Land M, respectively. If Mand Nare quasi-ideals of a Lie-Rinehart algebra L, then f:MˆNÑMXN, pm, nq ÞÑ rm, nsis a Lie-Rinehart pairing. We say that a Lie-Rinehart pairing f:LˆMÑLis universal if for any other Lie-Rinehart pairing f1:LˆMÑL1there is a unique Lie-Rinehart homomorphism ϕ:LÑL1making commutative the diagram: LˆMf,2 f1 #+ L ϕ L1. The Lie-Rinehart algebra Lis unique up to isomorphism which we will describe as the non-abelian tensor product of Land M. Definition 1.5.3. Let Land Mbe a pair of Lie-Rinehart algebras together with a quasi-action of Lon Mand a quasi-action of Mon L. We define the non-abelian tensor product of Land Min LRAK,LbM, as the Lie-Rinehart A-algebra spanned as an A-module by the symbols xbm, and subject only to the relations: 1. kapxbmq “ apkx bmq “ apxbkmq, 2. xb pm`nq “ xbm`xbn, px`yq b m“xbm`ybm, 3. rx, ys b m“xbym´ybxm, xb rm, ns “ nxbm´mxbn, 4. apmxq b bpynq “ abpmxbynq ´ arαLpxq, αMpmqspbqpybnq `brαLpyq, αMpnqspaqpxbmq, for every kPK,a, b PA,x, y PLand m, n PM, with the induced bracket rapxbmq, bpybnqs “ ´apmxq b bpynqand anchor map α:LbMÑDerKpAq given by α`apxbmq˘:“arαLpxq, αMpmqs.
1.5 Non-abelian tensor product of Lie-Rinehart algebras 33 This way, the map f:LˆMÑLbMwhich sends px, mqto xbmis a universal Lie-Rinehart pairing by construction. Definition 1.5.4. Two quasi-actions LˆMÑMand MˆLÑLare said to be compatible if for all x, y PLand m, n PM, 1. ´αLpmxq “ αMpxmq “ rαLpxq, αMpmqs, 2. pmxqn“ rn, xms, 3. pxmqy“ ry, mxs. This is the case, for example, if Land Mare both quasi-ideals of some Lie-Rinehart algebra and the quasi-actions are given by multiplication. We can see another example of compatible quasi-actions when B:LÑNand B1:MÑNare crossed modules. In this case, Land Mquasi-act on each other via the action of N. These quasi-actions are compatible. If A“Kthen we recover the notion of compatible actions between Lie algebras [6]. From this point on we shall assume that all quasi-actions are compatible. Proposition 1.5.5. Let µ:LbMÑLand ν:LbMÑMbe the homomorphisms defined on generators by µ`apxbmq˘“ ´apmxqand ν`apxbmq˘“ apxmq. They are Lie-Rinehart homomorphisms and the following diagram is commutative: LbMν,2 µ α $, M αM LαL,2DerKpAq. We can relate the Lie-Rinehart tensor product LbMwith the tensor product of Land Mas an A-module. We will denote it by Lb mod Mthe A-module generated by the symbols xbmsubject to the relations 1. kpxbmq “ kx bm“xbkm, 2. xb pm`nq “ xbm`xbn, px`yq b m“xbm`ybm, for every kPK,x, y PLand m, n PM.
34 1 Universal central extensions of Lie-Rinehart algebras Proposition 1.5.6. The canonical map Lb mod MÑLbMis a surjective A-module homomorphism. In addition, if Land Mquasi-act trivially on each other, there is an isomorphism of A-modules: LbM–L rL, Lsb mod M rM, Ms, where L{rL, Lsand M{rM, Msdenote the abelianization as K-Lie algebras. Proof. The identity (3) of Definition 1.5.3 is satisfied since the quasi-action is trivial and the identity (4) is consequence of this fact and of Definition 1.5.4 (1). Proposition 1.5.7. The Lie-Rinehart algebras LbMand MbLare isomorphic. Proof. The map f:LˆMÑMbLwhich sends px, mq Ñ mbxis a Lie-Rinehart pairing, then by the universal property of LbMthere is a LieRinehart homomorphism LbMÑMbL. In a similar way, we can construct the inverse MbLÑLbMand establish an isomorphism. Proposition 1.5.8. Consider the following short exact sequence of LieRinehart algebras Lf,2Mg,2N, and assume that there are compatible quasi-actions of a Lie-Rinehart algebra Pon L,Mand N, and of L, M, N on P. Suppose also that the Lie-Rinehart morphisms fand gpreserve these quasi-actions, i.e. fppxq “ pfpxq,xp“fpxqp and gppmq “ pgpmq,mp“gpmqp, where xPL,mPMand pPP. Then, the following sequence is exact LbPfb1,2MbPgb1,2NbP, with gb1surjective. Proof. Since fand gpreserve the quasi-actions, it is easy to see that fb1 and gb1are Lie-Rinehart algebra morphisms. Furthermore, the morphism gb1is clearly surjective, and Impfb1q Ă Kerpgb1q. Since gf “0, we have that fpxqpaq “ 0for every aPAand xPL. This means that pfb 1qpxbpqpaq “ rαM`fpxq˘, αPppqspaq “ 0. Moreover, Impfb1qis an Amodule and preserves the Lie bracket since fand gpreserve the quasi-actions,
1.5 Bibliography 35 so Impfb1qis an ideal. Then to prove the other inclusion, we will show that MbP{Impfb1q – NbP. Since Impfb1q Ă Kerpgb1qwe have a natural epimorphism φ:MbP{Impfb1q Ñ NbP. Now we define the map ψ:NˆPÑMbP{Impfb1qsuch that ϕpn, pq “ mbp`Impfb1qwhere m is such that gpmq “ n. It follows that it is a well-defined Lie-Rinehart pairing, so by the universality of the tensor product, there exists a unique Lie-Rinehart morphism ϕ:NbPÑMbP{Impfb1q, and it is straightforward that φand ϕare inverse morphisms. Proposition 1.5.9. Given a perfect Lie-Rinehart algebra L, the non-abelian tensor product LbLis the universal central extension of L, where the quasiaction of Lon itself is the Lie bracket. Proof. It is routine to check that LbLÑLis a central extension. To see the universality, given a central extension p:MÑL, we pick a section in Set, s:LÑM. We define now a map f:LˆLÑMby fpx, yq “ rspxq, spyqs. Doing the same trick as in Proposition 1.3.10, we see that is a Lie-Rinehart pairing, so it can be extended to LbLÑM. Since Lis perfect, we saw in Lemma 1.3.3 that this map is unique. Acknowledgments The first author was supported by Grant PIP 112-201101-00636-CONICETArgentina. The last two authors were supported by Agencia Estatal de Investigación (Spain), grant MTM2016-79661-P (European FEDER support included, UE) and by Grant of Xunta de Galicia GRC2013-045 (European FEDER support included, UE). The second author is also supported by an FPU scholarship, Ministerio de Educación, Cultura y Deporte (Spain). We thank the referee for the helpful comments and suggestions which made the paper more enhanced. Bibliography [1] D. Calaque, A PBW theorem for inclusions of (sheaves of) Lie algebroids, Rend. Semin. Mat. Univ. Padova 131 (2014), 23–47. [2] H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956.
42 2 Non-abelian tensor product of Lie superalgebras lowing identities: rx, ys “ ´p´1q|x||y|ry, xs, “x, ry, zs‰““rx, ys, z‰` p´1q|x||y|“y, rx, zs‰, rm¯ 0, m¯ 0s “ 0, for all homogeneous elements x, y, z PMand m¯ 0PM¯ 0. Note that the last equation is an immediate consequence of the first one in the case 2has an inverse in K. Moreover, it can be easily seen that the second equation is equivalent to the graded Jacobi identity p´1q|x||z|“x, ry, zs‰` p´1q|y||x|“y, rz, xs‰` p´1q|z||y|“z, rx, ys‰“0. For a Lie superalgebra M“M¯ 0‘M¯ 1, the even part M¯ 0is a Lie algebra. Hence, if M¯ 1“0, then Mis just a Lie algebra. A Lie superalgebra Mwithout even part, i. e., M¯ 0“0, is an abelian Lie superalgebra, that is, rx, ys “ 0for all x, y PM. ALie superalgebra homomorphism f:MÑM1is a supermodule homomorphism of even grade such that frx, ys “ rfpxq, fpyqs for all x, y PM. Example 2.2.2. (i) Any associative superalgebra Acan be considered as a Lie superalgebra with the bracket ra, bs “ ab ´ p´1q|a||b|ba. (ii) Let m,nbe positive integers and Aa unital associative superalgebra. Consider the algebra Mpm, n, Aqof all pm`nqˆpm`nq-matrices with entries in Aand with the usual product of matrices. A Z2-gradation is defined as follows: homogeneous elements are matrices Eijpaqhaving the homogeneous element aPAat the position pi, jqand zero elsewhere, and |Eijpaq| “ |i| ` |j| ` |a|, where |i| “ ¯ 0if 1ďiďmand |i| “ ¯ 1if m`1ďiďm`n. With this gradation, Mpm, n, Aqturns out to be an associative superalgebra. The corresponding Lie superalgebra will be denoted by glpm, n, Aq. (iii) Let V“V¯ 0‘V¯ 1be a supermodule. Then the supermodule EndKpVq of all linear endomorphisms VÑV(of both degrees 0and 1) has a structure of an associative superalgebra with respect to composition (see [2]) and hence
2.2 Preliminaries on Lie Superalgebras 43 becomes a Lie superalgebra. In particular, if the ground ring Kis a field, and m,nare dimensions of V¯ 0and V¯ 1respectively, then choosing a homogeneous basis of Vordered such that even elements stand before odd, the elements of EndKpVqcan be seen as pm`nq ˆ pm`nq-square matrices ˆa b c d˙ where a, b, c and dare respectively mˆm,mˆn,nˆmand nˆnmatrices with entries in K. The even elements are the matrices with b“c“0and the odd elements are matrices with a“d“0. Let Mand Nbe two submodules of a Lie superalgebra P. We denote by rM, Nsthe submodule of Pspanned by all elements rm, nswith mPMand nPN. A Z2-graded submodule Mis a graded ideal of Pif rM, P s Ď M. In particular, the submodule ZpPq “ tcPP:rc, ps “ 0for all pPPuis a graded ideal and it is called the centre of P. Clearly if Mand Nare graded ideals of P, then so is rM, Ns. Let Mbe a Lie superalgebra and DPEndKpMq. We say that Dis a derivation if for all x, y PM Dprx, ysq “ rDpxq, ys ` p´1q|D||x|rx, Dpyqs. We denote by `DerKpMq˘¯αthe set of homogeneous derivations of degree ¯αP Z2. One verifies that the supermodule of derivations DerKpMq “ `DerKpMq˘¯ 0‘`DerKpMq˘¯ 1 is a subalgebra of the Lie superalgebra EndKpMq. 2.2.2 Actions and crossed modules of Lie superalgebras Definition 2.2.3. Let Pand Mbe two Lie superalgebras. By an action of Pon Mwe mean a K-bilinear map of even grade, PˆMÑM, pp, mq ÞÑ pm, such that (i) rp,p1sm“ppp1 mq ´ p´1q|p||p1|p1ppmq,
44 2 Non-abelian tensor product of Lie superalgebras (ii) prm, m1s “ rpm, m1s ` p´1q|p||m|rm, pm1s, for all homogeneous p, p1PPand m, m1PM. The action is called trivial if pm“0for all pPPand mPM. For example, if Mis a graded ideal and Pis a subalgebra of a Lie superalgebra Q, then the bracket in Qinduces an action of Pon M. Note that the action of Pon Mis the same as a Lie superalgebra homomorphism PÑDerKpMq. Remark 2.2.4.If Mis an abelian Lie superalgebra enriched with an action of a Lie superalgebra P, then Mhas a structure of a supermodule over P (P-supermodule, for short) (see e. g. [18]), that is, there is a K-bilinear map of even grade PˆMÑM,pp, mq ÞÑ pm, such that rp, p1sm“ppp1mq ´ p´1q|p||p1|p1ppmq, for all homogeneous p, p1PPand mPM. Note that a P-supermodule Mis the same as a K-supermodule Mtogether with a Lie superalgebra homomorphism PÑEndKpMq. Definition 2.2.5. Given two Lie superalgebras Mand Pwith an action of Pon M, we can define the semidirect product M¸Pwith the underlying supermodule M‘Pendowed with the bracket given by rpm, pq,pm1, p1qs “ prm, m1s ` pm1´ p´1q|m||p1|pp1 mq,rp, p1sq. Now we are ready to introduce the following notion of crossed modules of Lie superalgebras (see also [23, Definition 5]). Definition 2.2.6. Acrossed module of Lie superalgebras is a homomorphism of Lie superalgebras B:MÑPwith an action of Pon Msatisfying (i) Bppmq “ rp, Bpmqs, (ii) Bpmqm1“ rm, m1s, for all pPPand m, m1PM.
2.2 Preliminaries on Lie Superalgebras 45 Example 2.2.7. There are some standard examples of crossed modules: (i) The inclusion MãÑPof a graded ideal Mof a Lie superalgebra Pis a crossed module of Lie superalgebras. (ii) If Pis a Lie superalgebra and Mis a P-supermodule, the trivial map 0: MÑPis a crossed module of Lie superalgebras. (iii) A central extension of Lie superalgebras B:MP(i.e., Ker B Ď ZpMq) is a crossed module of Lie superalgebras. Here the action of Pon Mis given by pm“ r ¯m, ms, where ¯mPMis any element of B´1ppq. (iv) The homomorphism of Lie superalgebras B:MÑDerKpMqwhich sends mPMto the inner derivation adpmq P DerKpMq, defined by adpmqpm1q “ rm, m1s, together with the action of DerKpMqon Mgiven by Dm“Dpmq, is a crossed module of Lie superalgebras. Lemma 2.2.8. Let B:MÑPbe a crossed module of Lie superalgebras. Then the following conditions are satisfied: (i) The kernel of Bis in the centre of M. (ii) The image of Bis a graded ideal of P. (iii) The Lie superalgebra Im Bacts trivially on the centre ZpMq, and so trivially on Ker B. Hence Ker Binherits an action of P{Im Bmaking Ker Ba P{Im B-supermodule. Proof. This is an immediate consequence of Definition 2.2.6. 2.2.3 Free Lie superalgebra and enveloping superalgebra of a Lie superalgebra Definition 2.2.9. The free Lie superalgebra on a Z2-graded set X“X¯ 0YX¯ 1 is a Lie superalgebra FpXqtogether with a degree zero map i:XÑFpXq such that if Mis any Lie superalgebra and j:XÑMis a degree zero map, then there is a unique Lie superalgebra homomorphism h: FpXq Ñ Mwith j“h˝i. The existence of free Lie superalgebras is guaranteed by an analogue of Witt’s theorem (see [18, Theorem 6.2.1]). In the sequel we need the following construction of the free Lie superalgebra.
46 2 Non-abelian tensor product of Lie superalgebras Construction 2.2.10. Let X“X¯ 0YX¯ 1be a Z2-graded set. Denote by magpXqthe free magma over the set X. The free superalgebra on X, denoted by algpXq, has as elements the finite sums řiλixi, where λiPKand xiare elements of magpXqand the multiplication in algpXqextends the multiplication in magpXq. Note that the grading is naturally defined in algpXq. The free Lie superalgebra FpXqis the quotient algpXq{I, where Iis the graded ideal generated by the elements xy ` p´1q|x||y|yx, p´1q|x||z|`xpyzq˘` p´1q|y||x|`ypzxq˘` p´1q|z||y|`zpxyq˘, x¯ 0x¯ 0, for all homogeneous x, y, z PXand x¯ 0PX¯ 0. Definition 2.2.11. The universal enveloping superalgebra of a Lie superalgebra Mis a pair pUpMq, σq, where UpMqis a unital associative superalgebra and σ:MÑUpMqis an even linear map satisfying σrx, ys “ σpxqσpyq ´ p´1q|x||y|σpyqσpxq,(2.2.1) for all homogeneous x, y PM, such that the following universal property holds: for any other pair pA, σ1q, where Ais a unital associative superalgebra and σ1:MÑAis an even linear map satisfying (2.2.1), there is a unique superalgebra homomorphism f: UpMq Ñ Asuch that f˝σ“σ1. Now we need to recall (see e. g. [21]) that, given two supermodules M and N, the tensor product of modules MbKNhas a natural supermodule structure with Z2-grading given by pMbKNq¯α“à ¯ β`¯γ“¯α pM¯ βbKN¯γq. In particular, the tensor power Mbn,ně2, has the induced Z2-grading. Hence the tensor algebra TpMqhas the Z2-grading extending that of M. We call TpMqthe tensor superalgebra. Construction 2.2.12. Let Mbe a Lie superalgebra and TpMqthe tensor superalgebra over the underlying supermodule of M. Consider the two-sided ideal JpMqof TpMqgenerated by all elements of the form mbm1´ p´1q|m||m1|m1bm´ rm, m1s,
2.2 Preliminaries on Lie Superalgebras 47 for all homogeneous m, m1PM. Then the quotient UpMq “ TpMq{JpMq is a unital associative superalgebra. By composing the canonical inclusion MÑTpMqwith the canonical projection TpMq Ñ UpMqwe get the canonical even linear map σ:MÑUpMq. Then the pair pUpMq, σqis the universal enveloping superalgebra of M(see [2]). Note that, as in the Lie algebra case, the universal enveloping superalgebra turns out to be a very useful tool for the representation theory of Lie superalgebras. In particular, by the universal property, it follows that a Lie supermodule over a Lie superalgebra Mis the same as a Z2-graded (left) UpMq-module (see [21, Chapter 1]). Let us consider Kwith Z2-grading concentrated in degree zero, that is, with K¯ 1“0. Then the trivial map from a Lie superalgebra Minto Kgives rise to a unique homomorphism of superalgebras ε: UpMq Ñ K. The kernel of ε, denoted by ΩpMq, is called the augmentation ideal of M. Obviously, ΩpMq is just the graded ideal of UpMqgenerated by σpMq. 2.2.4 Homology of Lie superalgebras Now we briefly recall from [18, 22] the definition of homology of Lie superalgebras. The Grassmann algebra of a Lie superalgebra P, denoted by ŹKpPq, is defined to be the quotient of the tensor superalgebra TpPqof Pby the ideal generated by the elements xby` p´1q|x||y|ybx, for all homogeneous x, y PP. Note that ŹKpPq “ Àną0Źn KpPq, where Źn KpPqis the image of Pbnin ŹKpPq, has an induced P-supermodule structure given by xpx1^ ¨ ¨ ¨ ^ xnq “ n ÿ i“1 p´1q|x|řkăi|xk|px1^ ¨ ¨ ¨ ^ rx, xis ^ ¨ ¨ ¨ ^ xnq. Let Mbe a P-supermodule and consider the chain complex pC˚pP, Mq, d˚q defined by CnpP, Mq “ Źn KpPq bKM, for ně0, with boundary maps
48 2 Non-abelian tensor product of Lie superalgebras dn:CnpP, Mq Ñ Cn´1pP, Mqdefined on generators by dnpx1^ ¨ ¨ ¨ ^ xnbyq “ n ÿ i“1 p´1qi`|xi|řkąi|xk|px1^ ¨ ¨ ¨ ^ ˆxi^ ¨ ¨ ¨ ^ xnbxiyq `ÿ iăj p´1qωprxi, xjs ^ ¨ ¨ ¨ ^ ˆxi^ ¨ ¨ ¨ ^ ˆxj^ ¨ ¨ ¨ ^ xnbyq, where ω“i`j`|xi|řkăi|xk|`|xj|řlăj|xl|`|xi||xj|. The n-th homology of the Lie superalgebra Pwith coefficients in the P-supermodule M,HnpP, Mq, is the n-th homology of the chain complex pC˚pP, Mq, d˚q, i.e. HnpP, Mq “ Ker dn Im dn`1 . If Kis regarded as a trivial P-supermodule, we write HnpPqfor HnpP, Kq. In the case when the ground ring Kis a field, there is a relation between Tor functor and the homology (see [18]) given by HnpP, Mq – TorUpPq npK, Mq. By analogy to Lie algebras (see e. g. [11]), we have the following isomorphisms H0pP, Mq – Coker `ΩpPq bUpPqMÝÑ M˘,(2.2.2) H1pP, Mq – Ker `ΩpPq bUpPqMÝÑ M˘.(2.2.3) 2.3 Non-abelian tensor product of Lie superalgebras In this section we introduce a non-abelian tensor product of Lie superalgebras, which generalizes the non-abelian tensor product of Lie algebras [8], and study its properties. 2.3.1 Construction of the non-abelian tensor product Definition 2.3.1. Let Mand Nbe two Lie superalgebras with actions on each other. Let XM,N be the Z2-graded set of all symbols mbn, where mPM¯ 0YM¯ 1,
2.3 Non-abelian tensor product of Lie superalgebras 49 nPN¯ 0YN¯ 1and the Z2-gradation is given by |mbn|“|m| ` |n|. We define the non-abelian tensor product of Mand N, denoted by MbN, as the Lie superalgebra generated by XM,N and subject to the relations: (i) λpmbnq “ λm bn“mbλn, (ii) pm`m1q b n“mbn`m1bn, where m, m1have the same grade, mb pn`n1q “ mbn`mbn1,where n, n1have the same grade, (iii) rm, m1s b n“mbm1 n´ p´1q|m||m1|pm1bmnq, mb rn, n1s “ p´1q|n1|p|m|`|n|qpn1 mbnq ´ p´1q|m||n|pnmbn1q, (iv) rmbn, m1bn1s “ ´p´1q|m||n|pnmbm1 n1q, for every λPK,m, m1PM¯ 0YM¯ 1and n, n1PN¯ 0YN¯ 1. Let us remark that if m“m¯ 0`m¯ 1is any element of Mand n“n¯ 0`n¯ 1 is any element of N, then under the notation mbnwe mean the sum m¯ 0bn¯ 0`m¯ 0bn¯ 1`m¯ 1bn¯ 0`m¯ 1bn¯ 1. If M“M¯ 0and N“N¯ 0then MbNis the non-abelian tensor product of Lie algebras introduced and studied in [8] (see also [12]). Definition 2.3.2. Actions of Lie superalgebras Mand Non each other are said to be compatible if (i) pnmqn1“ ´p´1q|m||n|rmn, n1s, (ii) pmnqm1“ ´p´1q|m||n|rnm, m1s, for all m, m1PM¯ 0YM¯ 1and n, n1PN¯ 0YN¯ 1. For example, if Mand Nare two graded ideals of some Lie superalgebra, the actions induced by the bracket are compatible. Proposition 2.3.3. Let Mand Nbe Lie superalgebras acting compatibly on each other. Then there is a natural isomorphism of Lie superalgebras MbN–MbKN DpM, Nq, where DpM, Nqis the submodule of the supermodule MbKNgenerated by the elements
50 2 Non-abelian tensor product of Lie superalgebras (i) rm, m1s b n´mbm1 n` p´1q|m||m1|pm1bmnq, (ii) mb rn, n1s ´ p´1q|n1|p|m|`|n|qpn1 mbnq ` p´1q|m||n|pnmbn1q, (iii) pnmq b pmnq, with |m|“|n| (iv) p´1q|m||n|pnmq b pm1 n1q ` p´1qp|m|`|n|qp|m1|`|n1|q`|m1||n1|pn1 m1qbpmnq, (v) ö pm,nq,pm1,n1q,pm2,n2q p´1qp|m|`|n|qp|m2|`|n2|q`|m||n|`|m1||n1|rnm, n1 m1sbpm2 n2q, for all m, m1, m2PM¯ 0YM¯ 1and n, n1, n2PN¯ 0YN¯ 1, where ö x,y,z denotes the cyclic summation with respect to x, y, z. Proof. There is a Lie superalgebra structure on the supermodule pMbK Nq{DpM, Nqgiven on generators by the following bracket rmbn, m1bn1s “ ´p´1q|m||n|pnmbm1 n1q, for all m, m1PM¯ 0YM¯ 1,n, n1PN¯ 0YN¯ 1and extended by linearity. It is routine to check that this bracket is compatible with the defining relations of pMbKNq{DpM, Nqand it indeed defines a Lie superalgebra structure. Then the canonical homomorphism MbNÑ pMbKNq{DpM, Nq,mbnÞÑ mbn, is an isomorphism. The proof of the following proposition is a routine calculation. Proposition 2.3.4. Let Mand Nbe two Lie superalgebras acting compatibly on each other. (i) The following morphisms µ:MbNÑM, m bnÞÑ ´p´1q|m||n|pnmq, ν:MbNÑN, m bnÞÑ mn, are Lie superalgebra homomorphisms. (ii) There are actions of Mand Non MbNgiven by m1pmbnq “ rm1, ms b n` p´1q|m||m1|mb pm1 nq, n1pmbnq “ pn1 mq b n` p´1q|n||n1|mb rn1, ns, for m, m1PM¯ 0YM¯ 1,n, n1PN¯ 0YN¯ 1and extended by linearity. Moreover, with these actions µand νare crossed modules of Lie superalgebras.
2.3 Non-abelian tensor product of Lie superalgebras 51 We will denote by rM, NsM(resp. rM, NsN) the image of µ(resp. ν), which by Lemma 2.2.8(ii) is a graded ideal of M(resp. N) generated by the elements of the form nm(resp. mn) for mPMand nPN. Note that by Lemma 2.2.8(iii) Kerpµq`resp. Kerpνq˘is an M{rM, NsM-supermodule (resp. N{rM, NsN-supermodule). 2.3.2 Some properties of the non-abelian tensor product The obvious analogues of Brown and Loday results [1] hold for Lie superalgebras. In the following two propositions immediately below we show that sometimes the non-abelian tensor product of Lie superalgebras can be expressed in terms of the tensor product of supermodules. Proposition 2.3.5. Let Mand Nbe Lie superalgebras acting on each other. Then the canonical map MbKNÑMbN,mbnÞÑ mbn, is an even, surjective homomorphism of supermodules. In addition, if Mand Nact trivially on each other, then MbNis an abelian Lie superalgebra and there is an isomorphism of supermodules MbN–Mab bKNab, where Mab “M{rM, Msand Nab “N{rN, Ns. Proof. It is straightforward by the identities (iv), (iii) of Definition 2.3.1. Proposition 2.3.6. Let Pbe a Lie superalgebra and MaP-supermodule considered as an abelian Lie superalgebra acting trivially on P. Then there is an isomorphism of supermodules PbM–ΩpPq bUpPqM. Proof. By Proposition 2.3.3 there is an isomorphism of supermodules PbM–PbKM W, where Wis the submodule of PbKMgenerated by all elements of the form rp, p1s b m´pbp1m` p´1q|p||p1|p1bpm
58 2 Non-abelian tensor product of Lie superalgebras Now the cyclic group Z{pn`1qZacts on AbKpn`1qvia tnpa0b ¨ ¨ ¨ b anq “ p´1qn`|an|řkăn|ak|anba0b ¨ ¨ ¨ b an´1, where tn“1` pn`1qZPZ{pn`1qZ. For each ně0, consider the quotient CnpAq “ AbKpn`1q{Imp1´tnqwhich is the module of coinvariants of C1 npAq under the Z{pn`1qZ-action. Then d1 ninduces a well-defined map dn:CnpAq Ñ Cn´1pAqand there is an induced chain complex pC˚pAq, d˚q, which is called the Connes complex of A. Its homologies are, by definition, the cyclic homologies of the associative superalgebra A, denoted by HCnpAq,ně0. Easy calculations show that, given an associative superalgebra A,HC1pAq is the kernel of the homomorphism of supermodules pAbKAq{ IpAq Ñ rA, As, a bbÞÑ ab ´ p´1q|a||b|ba, where rA, Asis the graded submodule of Agenerated by the elements ab ´ p´1q|a||b|ba and IpAqis the graded submodule of the supermodule AbKA generated by the elements abb` p´1q|a||b|bba, ab bc´abbc ` p´1q|c|p|a|`|b|qca bb, for all homogeneous a, b, c PA. Now let us consider Aas a Lie superalgebra `see Example 2.2.2(i)˘. Then there is a Lie superalgebra structure on pAbKAq{ IpAqgiven by rabb, a1bb1s “ ra, bsbra1, b1s for all a, a1, b, b1PA. We denote this Lie superalgebra by VpAq. In fact, VpAqis the quotient of the non-abelian tensor product AbAby the graded ideal generated by the elements xby` p´1q|x||y|ybxand xy bz´xbyz ` p´1q|z|p|x|`|y|qzx by, for all homogeneous x, y, z PA. Proposition 2.5.6. Let Abe a Lie superalgebra. Then the following assertions hold: (i) There are compatible actions of the Lie superalgebras Aand VpAqon each other. (ii) The map µ: VpAq Ñ Agiven by xbyÞÑ rx, ys, together with the action of Aon VpAq, is a crossed module of Lie superalgebras.
2.5 Non-abelian homology of Lie superalgebras 59 (iii) The action of Aon VpAqinduces the trivial action of Aon HC1pAq. (iv) There is a short exact sequence in the category CrosspAq 0Ñ pHC1pAq,0q Ñ pVpAq, µq Ñ prA, As, iq Ñ 0, where i:rA, As Ñ Ais the inclusion. Proof. (i) The action of Aon VpAqis induced by the action of Aon AbAgiven in Proposition 2.3.4(ii), that is apxbyq “ ra, xs b y` p´1q|a||x|xb ra, ys “ax by` p´1q|a|p|x|`|y|qxbya ´ p´1q|x||a|xbay ´ p´1q|x||a|xa by “abxy ´ p´1q|x||y|abyx “ab rx, ys, whilst the action of VpAqon Ais defined by xbya““rx, ys, a‰ for all homogeneous a, x, y PA. Straightforward calculations show that these are indeed (compatible) actions of Lie superalgebras. (ii) Since the crossed module of Lie superalgebras AbAÑA,xbyÞÑ rx, ys, given in Proposition 2.3.4, vanishes on the elements of the form xby` p´1q|x||y|ybxand xy bz´xbyz `p´1q|z|p|x|`|y|qzx by, then µis well defined and obviously it is a crossed module of Lie superalgebras. (iii) If řiλipxibyiq P HC1pAq, i.e. řiλirxi, yis “ 0, then for all aPAwe have a`ÿ i λipxibyiq˘“ÿ i λipab rxi, yisq “ abÿ i λirxi, yis “ 0. (iv) This is an immediate consequence of the assertions above.
60 2 Non-abelian tensor product of Lie superalgebras By Proposition 2.5.5 we have the following exact sequence of supermodules H1`A, HC1pAq˘H1`A, VpAq˘H1pA, rA, Asq H0`A, HC1pAq˘H0`A, VpAq˘H0pA, rA, Asq 0. (2.5.1) Below, we will calculate some of the terms of this exact sequence. At first, by analogy to the Dennis-Stein generators [6], we give a definition of the first Milnor cyclic homology for associative superalgebras. Definition 2.5.7. Let Abe an associative superalgebra. We define the first Milnor cyclic homology HCM 1pAqof Ato be the quotient of the supermodule AbKAby the graded ideal generated by the elements abb` p´1q|a||b|bba, ab bc´abbc ` p´1q|c|p|a|`|b|qca bb, abbc ´ p´1q|b||c|abcb, for all homogeneous a, b, c PA. It is clear that if Ais supercommutative, that is, ab “ p´1q|a||b|ba, for all homogeneous a, b PA, then HC1pAq – HCM 1pAq. Lemma 2.5.8. We have the following equalities and isomorphisms (i) H0`A, HC1pAq˘“HC1pAq, (ii) H1`A, HC1pAq˘–A{rA, As bKHC1pAq, (iii) H0pA, rA, Asq “ rA, As{“A, rA, As‰, (iv) H0`A, VpAq˘–HCM 1pAq. Proof. (i) Since Aacts trivially on HC1pAq, we have that Coker `AbHC1pAq Ñ HC1pAq˘“HC1pAq.
2.6 Non-abelian exterior product of Lie superalgebras 61 (ii) Since HC1pAqis abelian, by Proposition 2.3.5 we have that Ker `Ab HC1pAq Ñ HC1pAq˘–A{rA, As bKHC1pAq. (iii) and (iv) are straightforward. It follows that the exact sequence (2.5.1) can be written as in the following theorem. Theorem 2.5.9. If Ais a unital associative superalgebra. Then there is an exact sequence of supermodules A rA, AsbKHC1pAqH1`A, VpAq˘H1pA, rA, Asq HC1pAqHCM 1pAqrA, As “A, rA, As‰0. Corollary 2.5.10. If Ais perfect as a Lie superalgebra, we have an exact sequence 0ÑH1`A, VpAq˘ÑH2pAq Ñ HC1pAq Ñ 0, where H2pAqis the usual second homology of the Lie superalgebra A. If in addition H2pAq “ 0, then all terms of the exact sequence in the previous theorem are trivial. Proof. Since Ais perfect we know that H1pA, Aq – H2pAq,A{rA, As bK HC1pAq “ 0and the map AbVpAq Ñ VpAqis surjective. 2.6 Non-abelian exterior product of Lie superalgebras In this section we extend to Lie superalgebras the definition of the non-abelian exterior product of Lie algebras introduced in [8]. Then we use it to derive the Hopf formula for the second homology of a Lie superalgebra and to construct a six-term exact homology sequence of Lie superalgebras. 2.6.1 Construction of the non-abelian exterior product Let Pbe a Lie superalgebra and pM, Bq and pN, B1qtwo crossed P-modules. We consider the actions of Mand Non each other via P.
62 2 Non-abelian tensor product of Lie superalgebras Lemma 2.6.1. Let M˝Nbe the graded submodule of MbNgenerated by the elements (a) mbn` p´1q|m1||n1|m1bn1, where Bpmq “ B1pn1qand Bpm1q “ B1pnq, (b) m¯ 0bn¯ 0, where Bpm¯ 0q “ B1pn¯ 0q, with m, m1PM¯ 0YM¯ 1,n, n1PN¯ 0YN¯ 1,m¯ 0PM¯ 0and n¯ 0PN¯ 0. Then, M˝N is a graded ideal in the centre of MbN. Proof. Given an element mbn` p´1q|m1||n1|m1bn1of the form (a), suppose that |m1| “ |n|, then we have rxby, m bn` p´1q|m1||n1|m1bn1s “ ´p´1q|x||y|pyxq b `mn` p´1q|m1||n1|pm1 n1q˘ “ ´p´1q|x||y|pyxq b `Bpmqn` p´1q|m1||n1|pBpm1qn1q˘ “ ´p´1q|x||y|pyxq b `B1pn1qn` p´1q|m1||n1|pB1pnqn1q˘ “ ´p´1q|x||y|pyxq b `rn1, ns ` p´1q|n||n1|rn, n1s˘ “0. This is also true when |m1|‰|n|. Indeed, if |m1|‰|n|, since B,B1are even maps, the equality Bpmq “ B1pn1qholds if and only if Bpmq “ 0“ B1pn1q. Now take an element m¯ 0bn¯ 0of the form (b). Then we have rxby, m¯ 0bn¯ 0s “ ´p´1q|x||y|pyxqbpm¯ 0n¯ 0q “ ´p´1q|x||y|pyxqbpBpm¯ 0qn¯ 0q “ ´p´1q|x||y|pyxqbpBpn¯ 0qn¯ 0q “ ´p´1q|x||y|pyxqbrn¯ 0, n¯ 0s “0, for any xbyPMbN. This completes the proof. Definition 2.6.2. Let Pbe a Lie superalgebra and pM, Bq and pN, B1qtwo crossed P-modules. The non-abelian exterior product M^Nof the Lie superalgebras Mand Nis defined by M^N“MbN M˝N. The equivalence class of mbnwill be denoted by m^n.
2.6 Non-abelian exterior product of Lie superalgebras 63 Note that if M“M¯ 0and N“N¯ 0then M^Ncoincides with the nonabelian exterior product of Lie algebras [8]. Reviewing Section 2.3, one can easily check that most of results on the non-abelian tensor product are fulfilled for the non-abelian exterior product. In particular, there are homomorphisms of Lie superalgebras M^NÑM, M^NÑNand actions of Mand Non M^N, induced respectively by the homomorphisms and actions given in Proposition 2.3.4. It is also satisfied the isomorphism M^N–N^M. Further, given a short exact sequence of Lie superalgebras 0ÑKÑMÑPÑ0, as an exterior analogue of the exact sequence (2.3.2), we get the following exact sequence of Lie superalgebras K^MÑM^MÑP^PÑ0.(2.6.1) Given a Lie superalgebra M, since id:MÑMis a crossed module, we can consider M^M. It is the quotient of MbMby the following relations m^m1“ ´p´1q|m||m1|m1^m, m¯ 0^m¯ 0“0, for all m, m1PM¯ 0YM¯ 1and m¯ 0PM¯ 0. In the particular case when Mis perfect, it is easy to see that M˝M“0, so M^M–MbMand in Theorem 2.4.1 we can replace MbMby M^M. 2.6.2 A six term exact homology sequence In [7], the non-abelian exterior product of Lie algebras is used to construct a six-term exact sequence of homology of Lie algebras. In this section we will extend these results to Lie superalgebras. First of all, we prove an analogue of Miller’s theorem [17] on free Lie superalgebras extending the similar result obtained in [7] for Lie algebras. Proposition 2.6.3. Let F“FpXqbe the free Lie superalgebra on a graded set X. Then the homomorphism F^FÑF,x^yÞÑ xy is injective. Proof. Let us prove that rF, F s – F^F. Using the same notations as in Construction 2.2.10, we define a map φ: algpXq ˚ algpXq Ñ F^Fby řiλixiyiÞÑ řiλipxi^yiq, where algpXq ˚ algpXqis the free product of superalgebras. It is easy to see that φis a K-superalgebra homomorphism since rx^y, x1^y1s “ xy ^x1y1. The ideal Iis contained in algpXq ˚ algpXqand by
64 2 Non-abelian tensor product of Lie superalgebras using the defining relations of F^Fit is not difficult to check that φvanishes on I. So we have an induced map from rF, F sto F^F, which is inverse to the homomorphism F^FÑ rF, F s,x^yÞÑ xy. Let Pbe a Lie superalgebra and take the quotient supermodule pP^K Pq{ Im d3, where d3:Ź3 KpPq Ñ Ź2 KpPqis the boundary map in the homology complex pC˚pP, Kq, d˚q. Here Kis considered as a trivial P-module. We define a bracket in pP^KPq{ Im d3by setting rx^y, x1^y1s“rx, ys^rx1, y1s for all x, y PP. As a particular case of the exterior analogue of Proposition 2.3.3 we have Lemma 2.6.4. There is an isomorphism of Lie superalgebras P^KP Im d3 «P^P. Corollary 2.6.5. (i) For any Lie superalgebra Pthere is an isomorphism of supermodules H2pPq – KerpP^PÑPq. (ii) H2pFq “ 0if Fis a free Lie superalgebra. (iii) (Hopf Formula) Given a free presentation 0ÑRÑFÑPÑ0of a Lie superalgebra P, there is an isomorphism of supermodules H2pPq – RX rF, F s rF, Rs. Proof. (i) This follows immediately from Lemma 2.6.4. (ii) This is a consequence of (i) and Proposition 2.6.3. (iii) Since F^F– rF, F s, using the exact sequence (2.6.1), we have P^P–rF, F s rF, Rs. Then Lemma 2.6.4 completes the proof.
2.6 Bibliography 65 Theorem 2.6.6. Let Mbe a graded ideal of a Lie superalgebra P. Then there is an exact sequence KerpP^MÑPq Ñ H2pPq Ñ H2pP{Mq Ñ M rP, MsÑH1pPq Ñ H1pP{Mq Ñ 0. Proof. By using the exact sequence (2.6.1) we have the following commutative diagram of Lie superalgebras with exact rows M^P ,2P^P ,2P M^P M ,20 0,2M,2P,2P M,20. Since CokerpM^P–P^MÑMq – M{rP, Msand CokerpP^PÑ Pq – P{rP, P s – H1pPq, then the assertion follows by using snake lemma and Corollary 2.6.5(i). In particular, if Pis a Lie algebra and Mis an ideal of P, then this sequence coincides with the six-term exact sequence in the homology of Lie algebras obtained in [7]. Acknowledgements The authors wish to thank the anonymous referee for his help in improving the presentation of this paper. The authors were supported by Ministerio de Economía y Competitividad (Spain), grant MTM2013-43687-P (European FEDER support included). The first and third authors were also supported by Xunta de Galicia, grant GRC2013-045 (European FEDER support included). The first author was also supported by FPU scholarship, Ministerio de Educación, Cultura y Deporte (Spain). The second author was also supported by Xunta de Galicia, grant EM2013/016 (European FEDER support included) and Shota Rustaveli National Science Foundation, grant DI/12/5-103/11. Bibliography [1] R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), no. 3, 311–335, With an appendix by M. Zisman.
66 2 Non-abelian tensor product of Lie superalgebras [2] C. Carmeli, L. Caston, and R. Fioresi, Mathematical foundations of supersymmetry, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2011. [3] J. M. Casas and M. Ladra, Perfect crossed modules in Lie algebras, Comm. Algebra 23 (1995), no. 5, 1625–1644. [4] J. L. Castiglioni, X. García-Martínez, and M. Ladra, Universal central extensions of Lie-Rinehart algebras, J. Algebra Appl., 2017, doi:10.1142/S0219498818501347. [5] H. Chen and J. Sun, Universal central extensions of slm|nover Z{2Zgraded algebras, J. Pure Appl. Algebra 219 (2015), no. 9, 4278–4294. [6] R. K. Dennis and M. R. Stein, K2of discrete valuation rings, Advances in Math. 18 (1975), no. 2, 182–238. [7] G. J. Ellis, Nonabelian exterior products of Lie algebras and an exact sequence in the homology of Lie algebras, J. Pure Appl. Algebra 46 (1987), no. 2-3, 111–115. [8] G. J. Ellis, A nonabelian tensor product of Lie algebras, Glasgow Math. J. 33 (1991), no. 1, 101–120. [9] X. García-Martínez and M. Ladra, Universal central extensions of slpm, n, Aqof small rank over associative superalgebras, Turkish J. Math., 2017, doi:10.3906/mat-1604-3. [10] D. Guin, Cohomologie des algèbres de Lie croisées et K-théorie de Milnor additive, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 1, 93–118. [11] P. J. Hilton and U. Stammbach, A course in homological algebra, second ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. [12] N. Inassaridze, E. Khmaladze, and M. Ladra, Non-abelian homology of Lie algebras, Glasg. Math. J. 46 (2004), no. 2, 417–429. [13] K. Iohara and Y. Koga, Second homology of Lie superalgebras, Math. Nachr. 278 (2005), no. 9, 1041–1053.
2.6 Bibliography 67 [14] V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. [15] C. Kassel, A Künneth formula for the cyclic cohomology of Z{2-graded algebras, Math. Ann. 275 (1986), no. 4, 683–699. [16] C. Kassel and J.-L. Loday, Extensions centrales d’algèbres de Lie, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 119–142. [17] J. Milnor, Introduction to algebraic K-theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971, Annals of Mathematics Studies, No. 72. [18] I. M. Musson, Lie superalgebras and enveloping algebras, Graduate Studies in Mathematics, vol. 131, American Mathematical Society, Providence, RI, 2012. [19] E. Neher, An introduction to universal central extensions of Lie superalgebras, Groups, rings, Lie and Hopf algebras (St. John’s, NF, 2001), Math. Appl., vol. 555, Kluwer Acad. Publ., Dordrecht, 2003, pp. 141–166. [20] A. R. Salemkar, H. Tavallaee, H. Mohammadzadeh, and B. Edalatzadeh, On the non-abelian tensor product of Lie algebras, Linear Multilinear Algebra 58 (2010), no. 3-4, 333–341. [21] M. Scheunert, The theory of Lie superalgebras, Lecture Notes in Mathematics, vol. 716, Springer, Berlin, 1979. [22] J. Tanaka, On homology and cohomology of Lie superalgebras with coefficients in their finite-dimensional representations, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), no. 3, 51–53. [23] T. Zhang and Z. Liu, Omni-Lie superalgebras and Lie 2-superalgebras, Front. Math. China 9(2014), no. 5, 1195–2010.
74 3 Universal central extensions of slpm, n, Aq ψpx, yq “ ´p´1q|x||y|ψpy, xq, p´1q|x||z|ψprx, ys, zq ` p´1q|x||y|ψpry, zs, xq ` p´1q|y||z|ψprz, xs, yq “ 0, ψpx¯ 0, x¯ 0q “ 0, for all x, y, z PL,x¯ 0PL¯ 0. Given an even super 2-cocycle ψ, we can construct a central extension ([13]) L‘WÑL,px, wq ÞÑ x, where the bracket is given by rpx, w1q,py, w2qs “ `rx, ys, ψpx, yq˘(see [13]). In the particular case of L“stpm, n, Aqand the super 2-cocycle being surjective, this construction can be described in a different way using generators and relations. Definition 3.3.2. Let ψ:stpm, n, Aq ˆ stpm, n, Aq Ñ Wbe an even super 2cocycle, i.e. a super 2-cocycle such that |ψpx, yq| “ |x| ` |y|for homogeneous x, y Pstpm, n, Aq. Let stpm, n, Aq7be the Lie superalgebra generated by the elements Fijpaq7with homogeneous aPA,1ďi‰jďm`n, with degree |F7 ijpaq| “ |i|`|j|`|a|and by the elements of W, with the relations aÞÑ F7 ijpaqis a K-linear map, rW,Ws “ rF7 ijpaq,Ws “ 0, rF7 ijpaq, F7 jkpbqs “ F7 ikpabq ` ψ`Fijpaq, Fjkpbq˘for distinct i, j, k, rF7 ijpaq, F7 klpbqs “ ψ`Fijpaq, Fklpbq˘for i‰j‰k‰l‰i, where a, b PA. Lemma 3.3.3. If stpm, n, Aq1“stpm, n, Aq ‘ Wis a central extension constructed from a surjective super 2-cocycle ψ:stpm, n, Aq ˆ stpm, n, Aq Ñ W then there is an isomorphism ρ:stpm, n, Aq7Ñstpm, n, Aq1where ρ`F7 ijpaq˘“ Fij and ρpwq “ w. Proof. The proof of [3, Lemma 1] can be easily adapted. As before, we denote H7 ijpa, bq “ rF7 ijpaq, F7 jipbqs and h7pa, bq “ H7 1jpa, bq ´ p´1q|a||b|H7 1jp1, baq. Therefore, h7is independent of jand we have the analogue decomposition lemma.
3.4 Universal central extension of stp2,1, Aq75 Lemma 3.3.4. We can decompose the Lie superalgebra stpm, n, Aq7generated by a surjective super 2-cocycle ψ:stpm, n, Aq ˆ stpm, n, Aq Ñ Win the following way: stpm, n, Aq7“W‘h7pA, Aq ‘ ˆm`n à j“2 H7 1jp1, Aq˙à 1ďi‰jďm`n F7 ijpAq. 3.4 Universal central extension of stp2,1, Aq In this section we study the case when m`n“3and prove that stp2,1, Aqis the universal central extension of slp2,1, Aq. Theorem 3.4.1. If τ:r stp2,1, Aq Ñ stp2,1, Aqis a central extension, then there exists a unique section η:stp2,1, Aq Ñ r stp2,1, Aq. Proof. We will directly obtain a Lie superalgebra homomorphism η:stp2,1, Aq Ñ r stp2,1, Aq, such that τ˝η“id and since stp2,1, Aqis perfect it must be unique. Let 0,2V,2r stp2,1, Aqτ,2stp2,1, Aq,20 be a central extension. We choose a preimage for Fijpaqdenoted by r Fijpaq and extend it by K-linearity to all aPA. We define r Hijpa, bq “ r r Fijpaq,r Fjipbqs, since it is independent of the choice of r Fijpaq. By identity (3.2.4) we know that rr Hikp1,1q,r Fijpaqs “ r Fijpaq`vijpaq, where vijpaq P V, so we will replace r Fijpaqby r Fijpaq ` vijpaq. It suffices to show that these r Fijpaqsatisfy relations (3.2.1)–(3.2.3) because our K-linear section η:stp2,1, Aq Ñ r stp2,1, Aq,Fijpaq ÞÑ r Fijpaqwill be a Lie superalgebra homomorphism and the result is proved. The first relation is immediate by definition. To see the second one, we use Jacobi identity and the fact that Vis in the centre of r stp2,1, Aq. r Fijpabq “ r r Hikp1,1q,r Fijpabqs “ “r Hikp1,1q,rr Fikpaq,r Fkjpbqs‰ ““rr Hikp1,1q,r Fikpaqs,r Fkjpbq‰`“r Fikpaq,rr Hikp1,1q,r Fkjpbqs‰ “ r r Fikpa` p´1q|i|`|k|aq,r Fkjpbqs ` r r Fikpaq,´p´1qp|i|`|k|qp|i|`|k|q r Fkjpbqs “ r r Fikpaq,r Fkjpbqs.
76 3 Universal central extensions of slpm, n, Aq Now we check that the remaining brackets vanish. rr Fijpaq,r Fijpbqs “ “r Fijpaq,rr Fikpbq,r Fkjp1qs‰ ““rr Fijpaq,r Fikpbqs,r Fkjp1q‰ ` p´1qp|i|`|j|`|a|qp|i|`|k|`|b|q“r Fikpbq,rr Fijpaq,r Fkjp1qs‰“0. To see that rr Fijpaq,r Fikpbqs “ 0we can assume |i| ` |j| “ ¯ 1, then 0“ p´1q¯ 1`|a|“r Hijp1,1q,rr Fijpaq,r Fikpbqs‰ “ p´1q¯ 1`|a|“rr Hijp1,1q,r Fijpaqs,r Fikpbq‰ ` p´1qp¯ 1`|a|q`p|i|`|j|qp|i|`|j|`|a|q“r Fijpaq,rr Hijp1,1q,r Fikpbqs‰ “ p´1q¯ 1`|a|rr Fijpa` p´1q¯ 1aq,r Fikpbqs ` r r Fijpaq,r Fikpbqs “ r r Fijpaq,r Fikpbqs. If |i| ` |j| “ ¯ 0, we have that |i| ` |k| “ ¯ 1and the calculation is the same. Therefore, rr Fijpaq,r Fklpbqs “ 0if j‰kand i‰l, satisfying relation (3.2.3) and completing the proof. Corollary 3.4.2. The universal central extension of slp2,1, Aqand stp2,1, Aq is stp2,1, Aq. Moreover, H2`stp2,1, Aqq “ 0. 3.5 Universal central extension of stp3,1, Aq In this section we find the universal central extension of slp3,1, Aq. Let S4 be the symmetric group of degree 4, i.e. the set of all quadruples pi, j, k, lq where 1ďi, j, k, l ď4distinct. We quotient S4by Klein’s subgroup, formed by tp1,2,3,4q,p3,2,1,4q,p1,4,3,2q,p3,4,1,2qu, obtaining 6 cosets denoted by Pm. We have a map θthat sends pi, j, k, lq ÞÑ θ`pi, j, k, lq˘“mwhen pi, j, k, lq P Pm. Let ΠpA2qbe the K-supermodule A2(see Definition 3.2.3) with the parity changed, i.e., `ΠpA2q˘¯ 0“ pA2q¯ 1and `ΠpA2q˘¯ 1“ pA2q¯ 0. Let W“ΠpA2q6 be the K-supermodule formed by the direct sum of six copies of ΠpA2qand consider the maps m: ΠpA2q Ñ W,mp¯aq ÞÑ p0, . . . , ¯a, . . . , 0q, in the position m. Using the decomposition of Lemma 3.2.2 we consider the K-bilinear map ψ:stp3,1, Aq ˆ stp3,1, Aq Ñ W,
3.5 Universal central extension of stp3,1, Aq77 where ψ`Fijpaq, Fklpbq˘“θ`pi,j,k,lq˘pabq, ψpx, yq “ 0if xor ybelong to H. Lemma 3.5.1. The K-bilinear map ψis a super 2-cocycle. Proof. Since the grading in Wis changed and exactly one index is odd, we have that |ψ`Fijpaq, Fklpbq˘|“|i|`|j|`|a|`|k|`|l|`|b| “ |a|`|b|`¯ 1“ |θ`pi,j,k,lq˘pabq|, for homogeneous a, b PA, so ψis even. To complete the proof we can just follow the steps of [6, Lemma 2.2] since ¯a“ ´¯aso signs do not play any important role. By the previous lemma, we have a central extension 0,2W,2stp3,1, Aq7π,2stp3,1, Aq,20, where stp3,1, Aq7“stp3,1, Aq ‘ Wis the Lie superalgebra constructed by the surjective super 2-cocycle ψ, defined by the following relations aÞÑ F7 ijpaqis a K-linear map,(3.5.1) rW,Ws “ rF7 ijpaq,Ws “ 0,(3.5.2) rF7 ijpaq, F7 jkpbqs “ F7 ikpabqfor distinct i, j, k, (3.5.3) rF7 ijpaq, F7 ijpaqs “ 0,(3.5.4) rF7 ijpaq, F7 ikpbqs “ 0,(3.5.5) rF7 ijpaq, F7 klpbqs “ θ`pi,j,k,lq˘pabqfor distinct i, j, k, l. (3.5.6) Theorem 3.5.2. The central extension 0ÑWÑstp3,1, Aq7Ñstp3,1, Aqis universal. Proof. Let 0,2V,2r stp3,1, Aqτ,2stp3,1, Aq,20
78 3 Universal central extensions of slpm, n, Aq be a central extension. We need to show that there exists a Lie superalgebra homomorphism ρ:stp3,1, Aq7Ñr stp3,1, Aqsuch that τ˝ρ“π. We choose a preimage r Fijpaqof FijpaqK-linearly for all aPA. Since VĂZpr stp3,1, Aqq, we have that rr Fikpaq,r Fkjpbqs “ r Fijpabq ` vijkpa, bq, for distinct i, j, k, where vijkpa, bq P V. Using Jacobi identity we have rr Fikpaq,r Fkjpcbqs “ “r Fikpaq,rr Fklpcq,r Fljpbqs‰ ““rr Fikpaq,r Fklpcqs,r Fljpbq‰ ` p´1qp|i|`|k|`|a|qp|k|`|l|`|c|q“r Fklpcq,rr Fikpaq,r Fljpbqs‰ “ r r Filpacq,r Fljpbqs, so choosing c“1we have the identities vijkpa, bq “ vijlpa, bqand rr Fikpaq,r Fkjpbqs “ r r Filpaq,r Fljpbqs. This means that vijkpa, bqis independent of the choice of kso we have rr Fikpaq,r Fkjpbqs “ r Fijpabq ` vijpa, bq, and rr Fikp1q,r Fkjpbqs “ r Fijpbq ` vijp1, bq. Therefore, we can replace r Fijpbqby r Fijpbq ` vijp1, bq. We want to define ρ`F7 ijpaq˘“r Fijpaqso will see that these elements satisfy relations (3.5.1)– (3.5.6). Relations (3.5.1), (3.5.2) and (3.5.3) are straightforward by definition. To see relation (3.5.4), we choose i, j, k distinct rr Fijpaq,r Fijpbqs “ “r Fijpaq,rr Fikpbq,r Fkjp1qs‰ ““rr Fijpaq,r Fikpbqs,r Fkjp1q‰ ` p´1qp|i|`|j|`|a|qp|i|`|k|`|b|q“r Fikpbq,rr Fijpaq,r Fkjp1qs‰ “0.
3.5 Universal central extension of stp3,1, Aq79 For relation (3.5.5), taking i, j, k, l distinct, we have rr Fijpaq,r Fikpbqs “ “r Fijpaq,rr Filpbq,r Fikp1qs‰ ““rr Fijpaq,r Filpbqs,r Fikp1q‰ ` p´1qp|i|`|j|`|a|qp|i|`|l|`|b|q“r Filpbq,rr Fijpaq,r Fikp1qs‰ “0. To check relation (3.5.6) we define r Hijpa, bq “ r r Fijpaq,r Fjipbqs and following the steps of Lemma 3.2.1 we can check that for distinct i, j, k, l, r Hijpa, bq “ ´p´1qp|i|`|j|`|a|qp|i|`|j|`|b|q r Hjipb, aq, rr Hijpa, bq,r Fikpcqs “ r Fikpabcq, rr Hijpa, bq,r Fkipcqs “ ´p´1qp|a|`|b|qp|i|`|k|`|c|q r Fkipcabq, rr Hijpa, bq,r Fkjpcqs “ p´1qp|i|`|j|`|a|qp|i|`|j|`|b|q`p|a|`|b|qp|j|`|k|`|c|q r Fkjpcbaq, rr Hijpa, bq,r Fijpcqs “ r Fijpabc ` p´1qp|i|`|j|`|a||b|`|b||c|`|c||a|qcbaq, rr Hijpa, bq,r Fklpcqs “ 0. When i, j, k, l are distinct we denote rr Fijpaq,r Fklp1qs “ vijklpaq, where vijklpaq P V. We want that ρpθ`pi,j,k,lq˘pabqq “ vijklpabq, since rr Fijpaq,r Fklpbqs “ rρ`F7 ijpaq˘, ρ`F7 klpbq˘s “ρ`rF7 ijpaq, F7 klpbqsq “ ρpθ`pi,j,k,lq˘pabqq “ vijklpabq. Thus, we have to check that (R1) 2vijklpaq “ 0, (R2) vijklpaq “ vkjilpaq “ vilkjpaq “ vklij paq, (R3) vijklparb, csq “ 0, (R4) rr Fijpaq,r Fklpbqs “ vijklpabq.
80 3 Universal central extensions of slpm, n, Aq Assume |i|`|j| “ ¯ 0, 0““r Hijpa, bq,rr Fijpcq,r Fklp1qs‰ ““rr Hijpa, bq,r Fijpcqs,r Fklp1q‰´“r Fijpaq,rr Hijp1,1q,r Fklp1qs‰ “ r r Fijpabc ` p´1q|i|`|j|`|a||b|`|b||c|`|c||a|cbaq,r Fklp1qs “vijklpabc ` p´1q|a||b|`|b||c|`|c||a|cbaq. If b“c“1, we have that vijklp2aq “ 2vijklpaq “ 0, proving (R1). If c“1, we have that vijklpab ´ p´1q|a||b|baq “ 0, so 0“vijklpabc ` p´1q|a||b|`|b||c|`|c||a|cbaq “ vijkl`pab ` p´1q|a||b|baqc˘, implying (R2). If |k|`|l| “ ¯ 0, the calculation is the same. On the other hand, rr Fijpaq,r Fklpbqs “ “rr Fikpaq,r Fkjp1qs,r Fklpbq‰ ““r Fikpaq,rr Fkjp1q,r Fklpbqs‰ ´ p´1qp|i|`|k|`|a|qp|k|`|j|q“r Fkjp1q,rr Fikpaq,r Fklpbqs‰ “ p´1qp|l|`|k|`|b|qp|k|`|j|qrr Filpabq,r Fkjp1qs “vilkjpabq, since the sign does not play any role. Choosing b“1and using (R1), we have that vijklpaq “ vilkjpaq. Doing the same but changing the indexes we have relations (R3) and (R4). Thus, the morphism ρ:stp3,1, Aq7Ñr stp3,1, Aqdefined by ρ`F7 ijpaq˘“r Fijpaqand ρpθ`pi,j,k,lq˘pabqq “ vijklpabq is actually a Lie superalgebra homomorphism completing the proof. Corollary 3.5.3. The universal central extension of slp3,1, Aqis stp3,1, Aq7– slp3,1, Aq ‘ ΠpA2q6. Moreover, H2`stp3,1, Aq˘–W–ΠpA2q6.
3.6 Universal central extension of stp2,2, Aq81 3.6 Universal central extension of stp2,2, Aq In this section we find the universal central extension of stp2,2, Aq. As in the previous section we consider the partition of S4but with a small difference. Not all the cosets will be considered as equals. The coset formed by tp1,3,2,4q,p1,4,2,3q,p2,3,1,4q,p2,4,1,3qu, is named P5the one formed by tp3,1,4,2q,p3,2,4,1q,p4,1,3,2q,p4,2,3,1qu, is named P6. The order of the other cosets P1, . . . , P4, will not be relevant. Note that all the elements of P5and P6have the property that |i| “ |k|,|j| “ |l| and |i|`|j| “ |k|`|l| “ ¯ 1. Let σ:S4Ñ t´1,1ube a map defined by σ`pi, j, k, lq˘“1if pi, j, k, lq P P1, P2, P3or P4, in P5, σ`pi, j, k, lq˘“1if pi, j, k, lq“p1,3,2,4qor p2,4,1,3q, σ`pi, j, k, lq˘“ ´1if pi, j, k, lq “ p1,4,2,3qor p2,3,1,4q, and in P6, σ`pi, j, k, lq˘“1if pi, j, k, lq “ p3,1,4,2qor p4,2,3,1q, σ`pi, j, k, lq˘“ ´1if pi, j, k, lq “ p3,2,4,1qor p4,1,3,2q. Furthermore, let W“A4 2‘A2 0be K-supermodule formed by the direct sum of four copies of A2and two copies of A0and the maps mp¯aq “ p0,...,¯a, . . . , 0q in position m. Using the decomposition of Lemma 3.2.2 we consider the K-bilinear map ψ:stp2,2, Aq ˆ stp2,2, Aq Ñ W, where ψ`Fijpaq, Fklpbq˘“θ`pi,j,k,lq˘pabq,if pi, j, k, lq P P1, P2, P3, P4 ψ`Fijpaq, Fklpbq˘“ p´1q|b|σ`pi, j, k, lq˘θ`pi,j,k,lq˘pabq,if pi, j, k, lq P P5or P6, ψpx, yq “ 0if xor ybelong to H.
82 3 Universal central extensions of slpm, n, Aq Lemma 3.6.1. The K-bilinear map ψis a super 2-cocycle. Proof. The map is even since |i| ` |j| ` |k| ` |l| “ ¯ 0. To check antisymmetry, it suffices to see what happens when pi, j, k, lq P P5or P6since in the other cases the signs do not make any difference since A2and A0are commutative. Let pi, j, k, lq P P5, we know that |i|`|j| “ |k|`|l| “ ¯ 1, ´p´1q|Fij paq||Fklpbq|ψ`Fklpbq,Fijpaq˘“ ´p´1qp|i|`|j|`|a|qp|k|`|l|`|b|qψ`Fklpbq, Fijpaq˘ “ ´p´1qp¯ 1`|a|qp¯ 1`|b|qp´1q|a|σ`pk, l, i, jq˘5pbaq “ p´1q|b|`|a||b|σ`pk, l, i, jq˘5pp´1q|a||b|abq “ p´1q|b|σ`pi, j, k, lq˘θ`pi,j,k,lq˘pabq “ψ`Fijpaq, Fklpbq˘, since σ`pi, j, k, lq˘“σ`pk, l, i, jq˘and ab “ p´1q|a||b|ba. If pi, j, k, lqbelongs to P6it is analogue. The identity ψpx¯ 0, x¯ 0q “ 0where x¯ 0P`stp2,2, Aq˘¯ 0is straightforward by definition. The last step is to check Jacobi identity. In order to ease notation, we denote by Jpx, y, zqthe expression p´1q|x||z|ψprx, ys, zq ` p´1q|x||y|ψpry, zs, xq ` p´1q|y||z|ψprz, xs, yq. We have to check that Jpx, y, zq “ 0for all x, y, z Pstp2,2, Aq. Let ψprx, ys, zq ‰ 0. Using the decomposition of Lemma 3.2.2 we see that at most one of x, y belongs to H. We can assume that xPH. To exclude trivial cases we need that y“Fijpaqand z“Fklpbq, where i, j, k, l are distinct. If pi, j, k, lq P P1, . . . , P4, the signs does not make any difference so the proof is the same as in [6, Lemma 2.2]. Therefore, we just need to check when pi, j, k, lq “ p1,3,2,4q P P5since the other cases are similar. If x“hpc, dq, then Jpx, y, zq “ p´1qp|c|`|d|qp|b|`¯ 1qψ`rhpc, dq, F13paqs, F24pbq˘ ` p´1qp|a|`¯ 1qp|b|`¯ 1qψ`rF24, hpc, dqs, F13pbq˘ “ p´1qp|c|`|d|qp|b|`¯ 1qψ`F13`pab ´ p´1q|a||b|baqc˘, F24pbq˘`0 “ p´1qp|c|`|d|qp|b|`¯ 1q`|b|σ`p1,3,2,4q˘5`pab ´ p´1q|a||b|baqcb˘ “0.
3.6 Universal central extension of stp2,2, Aq83 If x“H12p1, cq, then Jpx, y, zq “ p´1q|c|p|a|`¯ 1qψ`rH12p1, cq, F13paqs, F24pbq˘ ` p´1qp|a|`¯ 1qp|b|`¯ 1qψ`rF24pbq, H12p1, cqs, F13paq˘ “ p´1q|c|p|a|`¯ 1qψ`F13pcaq, F24pbq˘ ` p´1qp|a|`¯ 1qp|b|`¯ 1q`|c|p|b|`¯ 1qψ`F24pcbq, F13paq˘ “ p´1q|c|p|b|`¯ 1q`|a|σ`p1,3,2,4q˘5pcabq ` p´1qp|a|`|c|`¯ 1qp|b|`¯ 1q`|b|σ`p2,4,1,3q˘5pcbaq “ p´1q|c|p|b|`¯ 1q`p´1q|a|5pcabq ` p´1qp|a|`¯ 1qp|b|`¯ 1q`|b|5pcbaq˘ “ p´1q|c|p|b|`¯ 1q`|a|`5pcab ´ p´1q|a||b|cbaq˘ “0. If x“H13p1, cq, then Jpx, y, zq “ p´1q|c|p|a|`¯ 1qψ`rH13p1, cq, F13paqs, F24pbq˘ “ψ`F13pca ` p´1q¯ 1`|a||c|acq, F24pbq˘ “ p´1q|b|σ`p1,3,2,4q˘5ppca ´ p´1q|a||c|acqbq “0. If x“H14p1, cq, then Jpx, y, zq “ p´1q|c|p|b|`¯ 1qψ`rH14p1, cq, F13paqs, F24pbq˘ ` p´1qp|a|`¯ 1qp|b|`¯ 1qψ`rF24pbq, H14p1, cqs, F13paq˘ “ p´1q|c|p|b|`¯ 1qψ`F13pcaq, F24pbq˘ ` p´1qp|a|`¯ 1qp|b|`¯ 1q`|c|ψ`F24pbcq, F13paq˘ “ p´1q|c|p|b|`¯ 1q`|b|σ`p1,3,2,4q˘5pcabq ` p´1qp|a|`¯ 1qp|b|`¯ 1q`|c|`|a|σ`p2,4,1,3q˘5pbcaq “ p´1q|b|`|c|`p´1q|c||b|5pcabq ´ p´1q|a||b|5pbcaq˘ “ p´1q|b|`|c|`p´1q|c||b|`|b||c|`|a||b|5pbcaq ´ p´1q|a||b|5pbcaq˘ “0.
90 3 Universal central extensions of slpm, n, Aq Furthermore, if it is in the kernel of ω, all the aimust be zero. Then the restriction of µto the kernel of d1is surjective. 3.8 Concluding remarks Combining the main theorems presented here with the main theorems of [3, 7] we have a complete characterization of H2`stpm, n, Aq˘and H2`slpm, n, Aq˘ for m`ně3. Theorem 3.8.1. Let Ka unital commutative ring and Aan associative unital K-superalgebra. Then, H2`stpm, n, Aq˘“ $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % 0for m`ně5or m“2, n “1, A6 3for m“3, n “0, A6 2for m“4, n “0, ΠpA2q6for m“3, n “1, A4 2‘A2 0for m“2, n “2, where Amis the quotient of Aby the ideal mA `ArA, As(Definition 3.2.3) and Πis the parity change functor. Theorem 3.8.2. Let Ka unital commutative ring and Aan associative unital K-superalgebra with a K-basis containing the identity. Then, H2`slpm, n, Aq˘“ $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % HC1pAqfor m`ně5or m“2, n “1, HC1pAq ‘ A6 3for m“3, n “0, HC1pAq ‘ A6 2for m“4, n “0, HC1pAq ‘ ΠpA2q6for m“3, n “1, HC1pAq ‘ A4 2‘A2 0for m“2, n “2, where Amis the quotient of Aby the ideal mA `ArA, As(Definition 3.2.3) and Πis the parity change functor. Acknowledgments The authors were supported by Ministerio de Economía y Competitividad (Spain), grant MTM2016-79661-P and by Xunta de Galicia, grant GRC2013045 (European FEDER support included). The first author was also supported
3.8 Bibliography 91 by FPU scholarship, Ministerio de Educación, Cultura y Deporte (Spain) and a Fundación Barrié scholarship. The authors would like to show their gratitude to the anonymous referee for his help in improving the presentation of this paper. Bibliography [1] P. A. Blaga, Dennis supertrace and the Hochschild homology of supermatrices, Russ. J. Math. Phys. 16 (2009), no. 3, 384–390. [2] S. Bloch, The dilogarithm and extensions of Lie algebras, Algebraic Ktheory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Math., vol. 854, Springer, Berlin-New York, 1981, pp. 1–23. [3] H. Chen and N. Guay, Central extensions of matrix Lie superalgebras over Z{2Z-graded algebras, Algebr. Represent. Theory 16 (2013), no. 2, 591–604. [4] G. J. Ellis, A nonabelian tensor product of Lie algebras, Glasgow Math. J. 33 (1991), no. 1, 101–120. [5] Y. Gao, Steinberg unitary Lie algebras and skew-dihedral homology, J. Algebra 179 (1996), no. 1, 261–304. [6] Y. Gao and S. Shang, Universal coverings of Steinberg Lie algebras of small characteristic, J. Algebra 311 (2007), no. 1, 216–230. [7] X. García-Martínez, E. Khmaladze, and M. Ladra, Non-abelian tensor product and homology of Lie superalgebras, J. Algebra 440 (2015), 464– 488. [8] H. Garland, The arithmetic theory of loop groups, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 5–136. [9] K. Iohara and Y. Koga, Second homology of Lie superalgebras, Math. Nachr. 278 (2005), no. 9, 1041–1053. [10] C. Kassel, A Künneth formula for the cyclic cohomology of Z{2-graded algebras, Math. Ann. 275 (1986), no. 4, 683–699.
92 3 Universal central extensions of slpm, n, Aq [11] C. Kassel and J.-L. Loday, Extensions centrales d’algèbres de Lie, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 119–142. [12] A. V. Mikhalev and I. A. Pinchuk, Universal central extensions of the matrix Lie superalgebras slpm, n, Aq, Combinatorial and computational algebra (Hong Kong, 1999), Contemp. Math., vol. 264, Amer. Math. Soc., Providence, RI, 2000, pp. 111–125. [13] E. Neher, An introduction to universal central extensions of Lie superalgebras, Groups, rings, Lie and Hopf algebras (St. John’s, NF, 2001), Math. Appl., vol. 555, Kluwer Acad. Publ., Dordrecht, 2003, pp. 141–166. [14] S. Shang, H. Chen, and Y. Gao, Central extensions of Steinberg Lie superalgebras of small rank, Comm. Algebra 35 (2007), no. 12, 4225–4244. [15] W. L. J. van der Kallen, Infinitesimally central extensions of Chevalley groups, Lecture Notes in Mathematics, Vol. 356, Springer-Verlag, Berlin, 1973.
Chapter 4 Universal central extensions of Leibniz superalgebras over superdialgebras Abstract We complete the problem of finding the universal central extension in the category of Leibniz superalgebras of slpm, n, Dqwhen m`ně3and Dis a superdialgebra, solving in particular the problem when Dis an associative algebra, superalgebra or dialgebra. To accomplish this task we use a different method than the standard studied in the literature. We introduce and use the non-abelian tensor square of Leibniz superalgebras and its relations with the universal central extension. Reference X. García Martínez and M. Ladra, Universal central extensions of Leibniz superalgebras over superdialgebras, Mediterr. J. Math. 14 (2017), no. 2, Art. 73, 15. 4.1 Introduction Leibniz algebras, the non-antisymmetric analogue of Lie algebras, were first defined by Bloh [1] and later recovered by Loday in [22] when he handled periodicity phenomena in algebraic K-theory. Many authors have studied 93
94 4 Universal central extensions of Leibniz superalgebras this structure and it has some interesting applications in Geometry and Physics ([16], [25], [6]). On the other hand, the theory of superalgebras arises directly from supersymmetry, a part of the theory of elemental particles, in order to have a better understanding of the geometrical structure of spacetime and to complete the substantial meaningful task of the unification of quantum theory and general relativity ([32]). The study of Lie or Leibniz superalgebras has been a very active field in the recent years since the classification of simple complex finite-dimensional Lie superalgebras by Kac in [14]. The study of central extensions is a very important topic in mathematics. There is a direct connection between central extensions and (co)homology, and they also have relations with Physics ([30]). In particular, universal central extensions have been studied in many different structures as groups [27], Lie algebras [11], [31] or Lie superalgebras [28]. A very interesting tool in the study of universal central extensions is the non-abelian tensor product introduced in [2] and extended to Lie algebras in [5] and to Lie superalgebras in [9]. The theory related with the universal central extension of the special linear algebra slpn, Aqhas been very active due its relation with cyclic homology and its relevance in algebraic K-theory. The first approach was in the category of Lie algebras by Kassel and Loday in [15] where they described it when ně5 and Ais an associative algebra, and in [8] it was obtained for ně3. For the Lie superalgebra slpn, Aqand Aan associative superalgebra, the universal central extension was given in [3]. For the special linear superalgebra slpm, n, Aq, the universal central extension for Aan associative algebra was found in [26] and [29]; and for Aan associative superalgebra was found in [4] and [10]. In the category of Leibniz algebras, the universal central extension of slpn, Aq(seen as a Leibniz algebra), where Ais an associative algebra, was found in [24] when ně5and in [13] when ně3. For the Leibniz superalgebra slpm, n, Aq, when m`ně5and Ais an associative algebra it was calculated in [21]. In [18] it was found for the Leibniz algebra slpm, Dqand for the Leibniz superalgebra slpm, n, Dq, where mě5and m`ně5, respectively, and Dis an associative dialgebra. The aim of this paper is to complete the task, finding the universal central extension of slpm, n, Dqwhere Dis a superdialgebra and m`ně3. Since associative algebras, associative superalgebras and dialgebras are all examples of associative superdialgebras, we will solve all cases at once. Moreover, we obtain a result contradicting a specific point of a theorem given in [18]. The most interesting part of this paper is that the method used is not the same
4.2 Preliminaries 95 as in all the papers cited above. Due its relation with central extensions, we introduce and use the non-abelian tensor square of Leibniz superalgebras providing another point of view to this topic. 4.2 Preliminaries Throughout the paper we fix a commutative ring Rwith unit. 4.2.1 Dialgebras We recall from [23] the definitions and basic examples of (super)dialgebras. Definition 4.2.1. An associative dialgebra (dialgebra for short) is an Rmodule equipped with two R-linear maps $:DbRDÑD, %:DbRDÑD, where $and %are associative and satisfy the following conditions: $ ’ & ’ % a% pb%cq “ a% pb$cq, pa$bq % c“a$ pb%cq, pa%bq $ c“ pa$bq $ c, for all a, b, c PD. An ideal IĂDis an R-submodule such that if aor bbelong to Ithen a%bPDand a$bPD. Abar-unit in Dis an element ePDsuch that for all aPD, a%e“a“e$a. Note that a bar-unit may not be unique. A unital dialgebra is a dialgebra with a chosen bar-unit, that will be denoted by 1. An associative superdialgebra (superdialgebra for short) is a dialgebra equipped with a Z-graded structure compatible with the two operations, i.e. D¯α$D¯ βĎD¯α`¯ βand D¯α%D¯ βĎD¯α`¯ β, for ¯α, ¯ βPZ. The concepts of bar-unit, unital and ideal are analogous in superdialgebras. Note that the bar-unit is always even.
96 4 Universal central extensions of Leibniz superalgebras Example 4.2.2. We introduce some examples of superdialgebras. (i) An associative (super)algebra defines a (super)dialgebra structure in a canonical way, where a%b“ab “a$b. If it is unital, then the superdialgebra is unital. (ii) Let pA, dqa differential associative (super)algebra, i.e., dpabq “ dpaqb` adpbqand d2“0. We define the two operations by a%b“adpbq, a$b“dpaqb. It is immediate to check that with these operations pA, dqis a (super)dialgebra. (iii) Let Aan associative (super)algebra, Man A-(super)bimodule and f:MÑAan A-(super)bimodule map. Then we can define a (super)dialgebra structure with operations m%m1“mfpm1q, m$m1“fpmqm1. (iv) Let Dand D1be two superdialgebras. The tensor product DbRD1is a superdialgebra where paba1q % pbbb1q “ p´1q|a1||b|pa%bqbpa1%b1q, paba1q $ pbbb1q “ p´1q|a1||b|pa$bqbpa1$b1q. 4.2.2 Leibniz superalgebras Definition 4.2.3. ALeibniz superalgebra Lis an R-supermodule with an R-linear even map r´,´s:LbRLÑL, satisfying the Leibniz identity “x, ry, zs‰““rx, ys, z‰´ p´1q|y||z|“rx, zs, y‰, for all x, y, z PL.
4.2 Preliminaries 97 Note that a Leibniz superalgebra where the identity rx, ys “ ´p´1q|x||y|ry, xsalso holds, is a Lie superalgebra. Example 4.2.4. (i) A Lie superalgebra is in particular a Leibniz superalgebra. (ii) Let Dbe a superdialgebra. Then Dwith the bracket ra, bs “ a%b´ p´1q|a||b|b$a, is a Leibniz superalgebra. If the two operations %and $are equal, i.e., Dis also an associative superalgebra, this bracket also induces a Lie superalgebra structure. Definition 4.2.5. The centre of a Leibniz superalgebra L, denoted by ZpLq, is the ideal formed by the elements zPLsuch that rz, xs“rx, zs “ 0for all xPL. The commutator of L, denoted by rL, Ls, is the ideal generated by the elements rx, yswhere x, y PL. A Leibniz superalgebra is called perfect if L“ rL, Ls. Definition 4.2.6. Acentral extension of a Leibniz superalgebra Lis a surjective homomorphism φ:MÑLsuch that Ker φĎZpMq. We say that a central extension u:UÑLis universal if for any central extension φ:MÑL there is a unique homomorphism f:UÑMsuch that u“φ˝f. The theory of central extensions of Leibniz superalgebras is studied in [20]. We obtain the following straightforward results. Proposition 4.2.7. Let φ:EÑMand ψ:MÑLbe two central extensions of Leibniz superalgebras. Then φis universal if and only if ψ˝φis universal. Proposition 4.2.8. Let Mbe a Leibniz superalgebra and Lan R-supermodule. An R-supermodule homomorphism ϕ:MÑLsuch that Ker ϕĎZpMqdefines a Leibniz superalgebra structure in Lwhere the bracket is given by rx, ys “ ϕprϕ´1pxq, ϕ´1pyqsq. for x, y PL. Now, we introduce the homology of Leibniz superalgebras with trivial coefficients adapting it from the non-graded version [24].
98 4 Universal central extensions of Leibniz superalgebras Definition 4.2.9. Let Lbe a Leibniz superalgebra and δn:LbnÑLbn´1 the R-linear map defined on generators by δnpx1b ¨ ¨ ¨ b xnq “ ÿ iăj p´1qn´j`|xj|p|xi`1|`¨¨¨`|xj´1|qx1b¨ ¨ ¨bxi´1brxi, xjsbxi`1b¨ ¨ ¨b pxjb¨ ¨ ¨bxn. The homology of Leibniz superalgebras with trivial coefficients is the homology of the chain complex formed by δn, i.e. HLnpLq “ Ker δn Im δn`1 Note that δ3pxbybzq “ ´rx, ys b z`xb ry, zs ` p´1q|y||z|rx, zs b y. In [12] it is defined a non-abelian tensor product of Leibniz algebras and in [17] a variation is introduced. Both coincide in the case of perfect Leibniz algebras, so for simplicity we will generalize to Leibniz superalgebras the version of [17]. Definition 4.2.10. Let Lbe a perfect Leibniz superalgebra. The non-abelian tensor product of Lis LbL“LbRL Im δ3 , where δ3is the map defined on the chain complex of Leibniz homology and the bracket is rxby, x1by1s “ rx, ys b rx1, y1s. Therefore, we have a short exact sequence. 0,2HL2pLq,2LbLδ2,2L,20. Theorem 4.2.11. Let Lbe a perfect Leibniz superalgebra. Then δ2:LbLÑL is the universal central extension of Land its kernel is HL2pLq. Proof. Let řixibyibe in the kernel of δ2. Then řirxi, yis “ 0, so rřixib yi, x1by1s “ řirxi, yisbrx1, y1s “ 0. Therefore, δ2is a central extension. Let 0,2Kι,2Mφ,2L,20be a central extension. We define a homomorphism u:LbLÑM,xbyÞÑ r¯x, ¯ys, where ¯xand ¯yare preimages by φof xand yrespectively. This homomorphism is well defined since Ker φĎ ZpMq. If u,u1are two homomorphisms such that φ˝u“φ˝u1, then u´u1“ι˝η where η:LbLÑKand ηprLbL, L bLsq “ 0. Since Lis perfect, LbLis also perfect and uis unique.
4.2 Preliminaries 99 4.2.3 Matrix Leibniz superalgebras Let t1, . . . , mu Y tm`1, . . . , m `nube a graded set and D“D¯ 0‘D¯ 1unital superdialgebra. We consider the set Mpm, n, Dqof pm`nqˆpm`nq-matrices. Let Eijpaqbe the matrix with aPDin the position pi, jqand zeros elsewhere. We define a grading in Mpm, n, Dqwhere the homogeneous elements are Eijpaq with ahomogeneous and the grading is given by |Eijpaq| “ |i| ` |j|`|a|. Now we define the general Leibniz superalgebra glpm, n, Dqwhich has Mpm, n, Dq with the previous grading as underlying set and the Leibniz bracket is given by rx, ys “ x%y´ p´1q|x||y|y$x. If m`ně2, we define the special linear Leibniz superalgebra slpm, n, Dq “ rglpm, n, Dq,glpm, n, Dqs. It is easy to see that slpm, n, Dqis generated by Eijpaqwith aPD¯ 0YD¯ 1and 1ďi‰jďm`n and the bracket is given by rEijpaq, Eklpbqs “ δjkEilpa%bq ´ p´1q|Eij paq||Eklpbq|δilEkjpb$aq. Following [18], if m`ně3then slpm, n, Dqis perfect. We define the supertrace as the R-linear homomorphism Str1:glpm, n, Dq Ñ Dwith Str1pxq “ m`n ÿ i“1 p´1q|i|p|i|`|xii|qxii. Note that slpm, n, Dq “ txPglpm, n, Dq: Str1pxqPrD, Dsu. Definition 4.2.12. Let Dbe a superdialgebra and mand nnon-negative integers such that m`ně3. We define the Steinberg Leibniz superalgebra denoted by stlpm, n, Dqas the Leibniz superalgebra generated by the elements Fijpaqwith aPD¯ 0YD¯ 1,1ďi‰jďm`n, where the grading is given by |Fijpaq| “ |i|`|j|`|a|, subject to the relations aÞÑ Fijpaqis R-linear, rFijpaq, Fklpbqs “ Filpa%bq, if i‰land j“k, rFijpaq, Fklpbqs “ ´p´1q|Fij paq||Fklpbq|Fkjpb$aq, if i“land j‰k, rFijpaq, Fklpbqs “ 0, if i‰land j‰k. We recall from [19] that stlpm, n, Dqis perfect and the canonical Leibniz superalgebra homomorphism φ:stlpm, n, Dq Ñ slpm, n, Dq,Fijpaq ÞÑ Eijpaq is a central extension.
106 4 Universal central extensions of Leibniz superalgebras and ϕpFijpaq b rFkipbq, Fijpcqsq “ vijkj pa%b%cq “ ´p´1q|a||b|vkjij `pb$aq % c˘ “ ´p´1q|a||b|ϕ`Fkjpb$aq b Fijpcq˘ “ϕ`rFijpaq, Fkipbqs b Fijpcq ´ p´1q|b||c|rFijpaq, Fijpcqs b Fkipbq˘. Now we define ψ:stlp3,0, Dq ‘ D6 3Ñstlp3,0, Dq b stlp3,0, Dqby Fijpaq ÞÑ Fikpaq b Fkj p1qand vijpqpaq ÞÑ Fijpaq b Fpqp1q. There is only one choice for k, but we need to check that it is well defined for elements of stlp3,0, Dq, since the arguments of Theorem 4.3.1 do not hold. Then, Fikpa%bq b Fkjp1q “ ´Fijpa%bqbrFkjp1q, Fjkp1qs “ ´rFikpaq, Fkjpbqs b rFkjp1q, Fjkp1qs “Fikpaq b `Fkjpbq ` Fkjpbq˘´Fikpaq b Fkjpbq “Fikpaq b Fkjpbq, and similarly for Fikpa%bq b Fjip1q “ Fikpaq b Fjipbq. Moreover, Fijpaq b Fijpbq “ Fijpaq b rFikpbq, Fkjp1qs “ 0. For the elements of D6 3, 0“ rFijpaq, Fikp1qs b rFikp1q, Fkip1qs “ Fijpaq b Fikp´3q “ Fijp3aq b Fikp1q, and 0“ rFijpaq, Fjipbqs b rFijpcq, Fikp1qs “Fijpa%b%c` p´1q|a||b|`|a||c|`|b||c|c$b$aq b Fikp1q ´ Fikpa%bq b Fijpcq, choosing b“c“1, Fikpaq b Fijp1q “ ´Fijpaq b Fikp1q, and choosing c“1, Fijp´a%b` p´1q|a||b|b$aq b Fikp1q “ 0.
4.5 Hochschild homology and Leibniz homology 107 To complete the proof, Fijpaq b Fikpbq “ ´Fijpaq b rFjkpbq, Fijp1qs “ ´Fikpa%bq b Fij p1q “Fijpa%bq b Fikp1q. 4.4.5 Case of slp2,1, Dq In this case, Wp2,1, Dq “ 0. Theorem 4.4.7. The universal central extension of slp2,1, Dqis stlp2,1, Dq. Proof. Defining the homomorphisms as in Theorem 4.4.6, we can recover the relations and additionally 0“ rFijpaq, Fikpbqs b rFikp1q, Fkip1qs “ Fijpaq b Fikp2` p´1q|i|`|k|1q. Therefore, if |i| “ ¯ 1or |k| “ ¯ 1, we have the relation Fijpaq b Fikp1q “ 0. If |j| “ ¯ 1, we do the same calculation but for Fikpaq b Fijp1q. It is similar for Fijpaq b Fkjp1q. 4.5 Hochschild homology and Leibniz homology In this section we adapt to the superalgebra case the definition of Hochschild homology of dialgebras introduced in [7] and we relate it with the universal central extension of slpm, n, Dq. From now on, we assume that Dis R-free. Let Dbe a superdialgebra with an R-basis containing the bar-unit. The boundary map dn:Dbn`1ÑDbnis defined on generators by dnpa0b ¨ ¨ ¨ b anq “ n´1 ÿ i“0 pa0b ¨ ¨ ¨ b ai%ai`1b ¨ ¨ ¨ b anq ` p´1qn`|an|řn´1 i“0|ai|pan$a0ba1b ¨ ¨ ¨ b an´1q, where aiPD. The Hochschild homology of superdialgebras, denoted by HH˚pDq, is the homology of the chain complex formed by the boundary maps d˚. Let Ibe the ideal of Dgenerated by the elements of the form abb%c´abb$c. We define HHS1pDq “ Ker d1 Im d2`I.
108 4 Universal central extensions of Leibniz superalgebras Theorem 4.5.1. There is an isomorphism of R-supermodules HL2`slpm, n, Dq˘–HHS1pDq ‘ Wpm, n, Dq. Proof. We have the following diagram 0,2HHS1pDq ‘ Wpm, n, Dq,2DbD Im d2`I‘Wpm, n, Dq µ d1,2rD, Ds E11p´q ,20 0,2HL2`slpm, n, Dq˘,2stlpm, n, Dq b stlpm, n, Dq Str2 LR ω,2slpm, n, Dq Str1 LR ,20, where µpabbq “ F1jpaq b Fj1pbq ´ p´1q|a||b|F1jpb$aq b Fj1p1q,µ`vijklpaq˘“ Fijpaq b Fklpbqand Str2`Fijpaq b Fklpbq˘$ ’ & ’ % abbif i“jand k“l vijklpa%bqif it makes sense depending on m, n 0otherwise. It is a straightforward computation that µ˝Str2and E11p´q˝Str1are the identity maps and that the diagram is commutative. Then the restriction of Str2 to the kernel of ωis also a split epimorphism, with µrestricted to the kernel of d1as section. Let us see that these restrictions are indeed isomorphisms. An element in the kernel of ωis a sum of elements of the form FijpaqbFjipbqplus elements of Wpm, n, Dq. Any element of Ker ωcan be written as an element of Im µplus řm`n i“2F1ipaiq b Fi1p1q, since Fijpaq b Fjipbq “ Fi1paq b F1ipbq ´ p´1qp|a|`|i|`|j|qp|b|`|i|`|j|qFj1pb$aq b F1jp1q, and F1jpaq b Fj1pbq “ F1jpaq b Fj1pbq ´ p´1q|a||b|Fj1pb$aq b F1jp1q ` p´1q|a||b|Fj1pb$aq b F1jp1q. Furthermore, if it is in the kernel of ω, all the aimust be zero. Then the restriction of µto the kernel of d1is surjective. Remark 4.5.2.The proof given in [21] can also be adapted since the assumptions on the characteristic of the ring are not used, but we rather give our version of the proof to show its relation with non-abelian tensor product.
4.6 Concluding remarks 109 4.6 Concluding remarks Combining the results obtained above we present the following summarizing theorems Theorem 4.6.1. Let Ra unital commutative ring and Dan associative unital R-superdialgebra with an R-basis containing the bar-unit. Then, HL2`slpm, n, Dq˘“ $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % HHS1pDqfor m`ně5or m“2, n “1, HHS1pDq ‘ D6 3for m“3, n “0, HHS1pDq ‘ D6 2for m“4, n “0, HHS1pDq ‘ ΠpD2q6for m“3, n “1, HHS1pDq ‘ D4 2‘D2 0for m“2, n “2, where Dmis the quotient of Dby the ideal mD ` prD, Ds % Dq(Definition 4.4.2) and Πis the parity change functor. Theorem 4.6.2. Let Ra unital commutative ring and Dan associative unital R-superdialgebra with an R-basis containing the bar-unit. Then, HL2`stlpm, n, Dq˘“ $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % 0for m`ně5or m“2, n “1, D6 3for m“3, n “0, D6 2for m“4, n “0, ΠpD2q6for m“3, n “1, D4 2‘D2 0for m“2, n “2, where Dmis the quotient of Dby the ideal mD ` prD, Ds % Dq(Definition 4.4.2) and Πis the parity change functor. Remark 4.6.3.We recall that in the case that m“n“2,Wp2,2, Dqmight not be zero even if charpRq ‰ 2contradicting [18, Theorem 6.2]. Acknowledgments The authors were supported by Ministerio de Economía y Competitividad (Spain), grants MTM2013-43687-P and MTM2016-79661-P(European FEDER support included) and by Xunta de Galicia, grant GRC2013-045 (European FEDER support included). The first author was also supported by
110 4 Universal central extensions of Leibniz superalgebras FPU scholarship, Ministerio de Educación, Cultura y Deporte (Spain) and a Fundación Barrié scolarship. We thank the referee for the helpful comments and suggestions which made the paper more enhanced. Bibliography [1] A. Bloh, On a generalization of the concept of Lie algebra, Dokl. Akad. Nauk SSSR 165 (1965), 471–473. [2] R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), no. 3, 311–335, With an appendix by M. Zisman. [3] H. Chen and N. Guay, Central extensions of matrix Lie superalgebras over Z{2Z-graded algebras, Algebr. Represent. Theory 16 (2013), no. 2, 591–604. [4] H. Chen and J. Sun, Universal central extensions of slm|nover Z{2Zgraded algebras, J. Pure Appl. Algebra 219 (2015), no. 9, 4278–4294. [5] G. J. Ellis, A nonabelian tensor product of Lie algebras, Glasgow Math. J. 33 (1991), no. 1, 101–120. [6] R. Felipe, N. López-Reyes, and F. Ongay, R-matrices for Leibniz algebras, Lett. Math. Phys. 63 (2003), no. 2, 157–164. [7] A. Frabetti, Leibniz homology of dialgebras of matrices, J. Pure Appl. Algebra 129 (1998), no. 2, 123–141. [8] Y. Gao and S. Shang, Universal coverings of Steinberg Lie algebras of small characteristic, J. Algebra 311 (2007), no. 1, 216–230. [9] X. García-Martínez, E. Khmaladze, and M. Ladra, Non-abelian tensor product and homology of Lie superalgebras, J. Algebra 440 (2015), 464– 488. [10] X. García-Martínez and M. Ladra, Universal central extensions of slpm, n, Aqof small rank over associative superalgebras, Turkish J. Math., 2017, doi:10.3906/mat-1604-3.
4.6 Bibliography 111 [11] H. Garland, The arithmetic theory of loop groups, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 5–136. [12] A. V. Gnedbaye, A non-abelian tensor product of Leibniz algebras, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 4, 1149–1177. [13] Q. Jiang, R. Shen, and Y. Su, Second homology groups and universal coverings of Steinberg Leibniz algebras of small characteristic, Comm. Algebra 37 (2009), no. 2, 548–566. [14] V. G. Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96. [15] C. Kassel and J.-L. Loday, Extensions centrales d’algèbres de Lie, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 119–142. [16] M. K. Kinyon and A. Weinstein, Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. J. Math. 123 (2001), no. 3, 525–550. [17] R. Kurdiani and T. Pirashvili, A Leibniz algebra structure on the second tensor power, J. Lie Theory 12 (2002), no. 2, 583–596. [18] D. Liu, Steinberg-Leibniz algebras and superalgebras, J. Algebra 283 (2005), no. 1, 199–221. [19] D. Liu and N. Hu, Steinberg unitary Liebniz algebras, Linear Algebra Appl. 405 (2005), 279–303. [20] D. Liu and N. Hu, Leibniz superalgebras and central extensions, J. Algebra Appl. 5(2006), no. 6, 765–780. [21] D. Liu and N. Hu, Universal central extensions of the matrix Leibniz superalgebras slpm, n, Aq, Comm. Algebra 35 (2007), no. 6, 1814–1823. [22] J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. (2) 39 (1993), no. 3-4, 269–293. [23] J.-L. Loday, Dialgebras, Dialgebras and related operads, Lecture Notes in Math., vol. 1763, Springer, Berlin, 2001, pp. 7–66. [24] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), no. 1, 139–158.
112 4 Universal central extensions of Leibniz superalgebras [25] J. M. Lodder, Leibniz cohomology for differentiable manifolds, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 73–95. [26] A. V. Mikhalev and I. A. Pinchuk, Universal central extensions of the matrix Lie superalgebras slpm, n, Aq, Combinatorial and computational algebra (Hong Kong, 1999), Contemp. Math., vol. 264, Amer. Math. Soc., Providence, RI, 2000, pp. 111–125. [27] J. Milnor, Introduction to algebraic K-theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971, Annals of Mathematics Studies, No. 72. [28] E. Neher, An introduction to universal central extensions of Lie superalgebras, Groups, rings, Lie and Hopf algebras (St. John’s, NF, 2001), Math. Appl., vol. 555, Kluwer Acad. Publ., Dordrecht, 2003, pp. 141–166. [29] S. Shang, H. Chen, and Y. Gao, Central extensions of Steinberg Lie superalgebras of small rank, Comm. Algebra 35 (2007), no. 12, 4225–4244. [30] G. M. Tuynman and W. A. J. J. Wiegerinck, Central extensions and physics, J. Geom. Phys. 4(1987), no. 2, 207–258. [31] W. L. J. van der Kallen, Infinitesimally central extensions of Chevalley groups, Lecture Notes in Mathematics, Vol. 356, Springer-Verlag, Berlin, 1973. [32] V. S. Varadarajan, Supersymmetry for mathematicians: an introduction, Courant Lecture Notes in Mathematics, vol. 11, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2004.
Chapter 5 A non-abelian exterior product and homology of Leibniz algebras Abstract We introduce a non-abelian exterior product of two crossed modules of Leibniz algebras and investigate its relation to the low-dimensional Leibniz homology. Later this non-abelian exterior product is applied to the construction of an eight term exact sequence in Leibniz homology. Also its relationship to the universal quadratic functor is established, which is applied to the comparison of the second Lie and Leibniz homologies of a Lie algebra. Reference G. Donadze, X. García-Martínez, and E. Khmaladze, A non-abelian exterior product and homology of Leibniz algebras, Rev. Mat. Complut., 2017, doi:10.1007/s13163-017-0237-2. 5.1 Introduction Leibniz algebras were first defined in 1965 by Bloh [2] as a non skew-symmetric analogue of Lie algebras but they became very popular when in 1993 Loday rediscovered them in [22]. One of the main reasons that Loday had to introduce 113
114 5 A non-abelian exterior product of Leibniz algebras them was that in the Lie homology complex the only property of the bracket needed was the so called Leibniz identity. Therefore, one can think about this notion as “non-commutative” analog of Lie algebras and study its homology. Since then, many authors have been studying them obtaining very relevant algebraic results ([24], [25]) and due their relations with Physics ([20], [26]) and Geometry ([17]). Many results of Lie algebras have been extended to the Leibniz case. As an example of these generalizations, Gnedbaye [18] extended to Leibniz algebras the notion of non-abelian tensor product, defined by Brown and Loday in the context of groups [5] and by Ellis in the context of Lie algebras [15]. The non-abelian tensor product was firstly introduced as a tool in homotopy theory, but it can give us nice information about central extensions and (co)homology. The notion of non-abelian exterior product was introduced in groups by Brown and Loday [4] and also extended to the Lie case by Ellis [15, 14]. The main objective of this manuscript is to give a proper generalization of this concept to Leibniz algebras. Given two ideals aand bof a Lie algebra g, the non-abelian exterior product is the quotient of the non-abelian tensor product a‹bby the elements of the form c˚c, where cPaXb. This makes sense in Lie theory since these are elements of the kernel of the homomorphism g‹gÑg, g˚g1ÞÑ rg, g1sfor all g, g1Pg, but this is not true in Leibniz algebras due to the lack of antisymmetry. Nevertheless, the non-abelian tensor product of two ideals of a Leibniz algebra, has some duplicity in the elements of the intersection, so this will be the way we can affront the problem. In fact, our definition of the non-abelian exterior product is given for crossed modules of Leibniz algebras, which is more general concept than ideals of Leibniz algebras. The paper is organized as follows. In Section 5.2 we recall some basic definitions and properties of Leibniz algebras and Leibniz homology. In Section 5.3 we introduce the non-abelian exterior product and we study its connections with the second Leibniz homology. In Section 5.4 we obtain an eight term exact sequence using the non-abelian exterior product. In Section 5.5 we explore the relations with Whitehead’s universal quadratic functor. Finally, in Section 5.6 we compare the second Leibniz homology and the second Lie homology of a Lie algebra.
5.2 Leibniz algebras and homology 115 5.2 Leibniz algebras and homology Throughout the paper Kis a field, unless otherwise stated. All vector spaces and algebras are considered over K, all linear maps are K-linear maps and b stands for bK. Definition 5.2.1 ([22]).A Leibniz algebra is a vector space gequipped with a linear map (Leibniz bracket) r,s:gbgÑg satisfying the Leibniz identity rx, ry, zss “ rrx, ys, zs ´ rrx, zs, ys, for all x, y, z Pg. A homomorphism of Leibniz algebras is a linear map preserving the bracket. The respective category of Leibniz algebras will be denoted by Lb. A subspace aof a Leibniz algebra gis called (two-sided) ideal of gif ra, xs,rx, as P afor all aPaand xPg. In this case the quotient space g{a naturally inherits a Leibniz algebra structure. An example of ideal of a Leibniz algebra gis the commutator of g, denoted by rg,gs, which is the subspace of gspanned by elements of the form rx, ys, x, y Pg. The quotient g{rg,gsis denoted by gab and is called abelianization of g. One more example of an ideal is the center Cpgq “ tcPg| rx, cs “ 0“ rc, xs,for all xPguof g. Note that both gab and Cpgqare abelian Leibniz algebras, that is, Leibniz algebras with the trivial Leibniz bracket r,s “ 0. Clearly, any Lie algebra is a Leibniz algebra and conversely, any Leibniz algebra with the antisymmetric Leibniz bracket is a Lie algebra. This is why Leibniz algebras are called non-commutative generalization of Lie algebras. Thus, there is a full embedding functor Lie ãÑLb, where Lie denotes the category of Lie algebras. This embedding has a left adjoint Lie:Lb ÑLie, called the Liezation functor and defined as follows. Given a Leibniz algebra g,Liepgqis the quotient of gby the subspace (that automatically is an ideal) spanned by elements of the form rx, xs,xPg(see for example [21]). The original reason to introduce Leibniz algebras was a new variant of Lie homology, called non-commutative Lie homology or Leibniz homology, developed in [23, 24] and denoted by HL˚. Let us recall the definition of HL˚.
122 5 A non-abelian exterior product of Leibniz algebras by the elements of the form i1pcq N i2pc1q ´ i2pcq N i1pc1q, where c, c1PaXb; i1:aXbÑaand i2:aXbÑbare the natural inclusions. The proof of the following proposition is immediate. Proposition 5.3.7. Let aand bbe two ideals of a Leibniz algebra. There is a homomorphism of Leibniz algebras θa,b:aNbÑaXb defined on generators by θa,bpaNbq “ ra, bsand θa,bpbNaq “ rb, as, for all aPaand bPb. Moreover, θa,bis a crossed module of Leibniz algebras (c.f. [18, Proposition 4.3]). Proposition 5.3.8. Let gbe a perfect Leibniz algebra, that is, g“ rg,gs. Then g‹g“gNgand the homomorphism θg,g:gNggis the universal central extension of g. Proof. The last four identities of the non-abelian tensor product immediately imply that g‹g“gNg. Hence, by [18, Theorem 6.5] the homomorphism θg,g:gNggis the universal central extension of the perfect Leibniz algebra g. 5.3.4 Relationship to the second homology Let γ:gÑhbe a homomorphism of Leibniz algebras, aand a1(resp. band b1) be two ideals of g(resp. h) such that γpaq Ď band γpa1q Ď b1. Since γpaXa1q Ď bXb1, it is easy to see that γinduces a homomorphism of Leibniz algebras aNa1ÑbNb1in the natural way: aNa1ÞÑ γpaq N γpa1qand a1NaÞÑ γpa1q N γpaq, for all aPa,a1Pa1. Now suppose that 0ÑaÑgÑhÑ0and 0Ña1Ñg1Ñh1Ñ0are extensions of Leibniz algebras, where a1and g1are ideals of g, while h1is an ideal of h. Then the following naturally induced map gNa1ˆaNg1ÑgNg1is not in general a homomorphism of Leibniz algebras, but the following sequence gNa1ˆaNg1ÑgNg1ÑhNh1Ñ0(5.3.1) is exact, in the sense that Im `gNa1ˆaNg1ÑgNg1˘“Ker `gNg1ÑhNh1˘. Lemma 5.3.9. Let 0ÑaÑgÑhÑ0be an extension of Leibniz algebras. Then, the following induced sequence of Leibniz algebras aNgÑgNgÑ hNhÑ0is exact.
5.3 Non-abelian tensor and exterior product of Leibniz algebras 123 Proof. Since the images of the induced homomorphisms aNgÑgNgand gNaÑgNgare the same, the statement follows immediately from the exactness of (5.3.1). Proposition 5.3.10. Let fbe a free Leibniz algebra over a set X. Then θf,f is injective. Proof. We will consider the epimorphism θf,f:fNfrf,fsand show that it is an isomorphism. Using the same notations as in Subsection 5.2.1, suppose rMX, MXsdenotes the subset trx, ys | x, y PMXuof MXand KrMX, MXs denotes the free vector space over the set rMX, MXs. Then, rf,fs “ KrMX, MXs NpSXq. Note that for each element mP rMX, MXsthere are unique xand yin MX such that m“ rx, ys. Therefore, the following map τ:KrMX, MXs Ñ fNf, given by rx, ys ÞÑ xNyfor each x, y PMpXq, is well defined. We have rx, ry, zss ´ rrx, ys, zs ` rrx, zs, ysτ ÞÑ xN ry, zs´rx, ys N z` rx, zs N y“0, for each x, y, z PMX. Moreover, if m“ n ř i“1 kixiPSXwith k1, . . . , knPKand x1, . . . , xnPMX, then we have: rx, msτ ÞÑ n ÿ i“1 kipxNxiq “ xN´n ÿ i“1 kixi¯“0, rm, xsτ ÞÑ n ÿ i“1 kipxiNxq “ ´n ÿ i“1 kixi¯Nx“0, for each xPMX. As a result we have that τpNpSXqq “ 0. Thus, τinduces a well-defined linear map τ˚:rf,fs Ñ fNf. Furthermore, τ˚˝θf,f“1fNfand θf,f˝τ˚“1rf,fs. This completes the proof. Corollary 5.3.11. Let 0ÑrÑfÑgÑ0be a free presentation of a Leibniz algebra g. Then there is an isomorphism gNg– rf,fs{rr,fs. Proof. This follows from Lemma 5.3.9 and Proposition 5.3.10.
124 5 A non-abelian exterior product of Leibniz algebras Theorem 5.3.12. Let gbe a Leibniz algebra. Then there is an isomorphism HL2pgq – Ker ´θg,g:gNgÑg¯. Proof. Let 0ÑrÑfÑgÑ0be a free presentation of g. By the Hopf formula we have HL2pgq – Ker ´rf,fs{rr,fs Ñ g¯. Thus, Corollary 5.3.11 completes the proof. Proposition 5.3.13. Let 0ÑaÑgÑhÑ0be an extension of Leibniz algebras. Then we have the following exact sequence Ker `θa,g:aNgÑa˘ÑHL2pgq Ñ HL2phq Ña{ra,gs Ñ HL1pgq Ñ HL1phq Ñ 0. Proof. By Lemma 5.3.9 we have the following commutative diagram with exact rows aNg,2 θa,g gNg,2 θg,g hNh,2 θh,h 0 0,2a,2g,2h,20. Now the Snake Lemma and Theorem 5.3.12 yield the exact sequence. Let g‚gdenote the vector space Cokerpgbgbgd Ñgbgq, where dis the boundary map in CL˚pgq. Let δ:g‚gÑgNgbe a linear map given by x‚yÞÑ xNy, where x‚ydenotes the coset of xbyPgbginto g‚g. It is easy to check that δis well defined. Proposition 5.3.14. The linear map δ:g‚gÑgNgis an isomorphism of vector spaces. Proof. We have the following commutative diagram with exact rows:
5.4 Third homology and the eight term exact sequence 125 0,2Ker d1,2 g‚gd1,2 δ g,2g{rg,gs 0,2Ker θg,g,2gNgθg,g,2g,2g{rg,gs, where d1is given by x‚yÞÑ rx, ysfor each x, y Pg. Since HL2pgq “ Ker d1, by Theorem 5.3.12 we have an isomorphism Ker d1–Ker θg,g. It is easy to verify that this isomorphism is induced by δ. Hence, the above diagram proves the proposition. Remark 5.3.15.It is shown in [21] that the vector space g‚ghas the Leibniz algebra structure given by rx‚y, x1‚y1s“rx, ys‚rx1, y1s, for each x, y, x1, y1Pg. This fact results from the previous proposition, because in gNgwe have that rxNy, x1Ny1s “ rx, ysNrx1, y1s. 5.4 Third homology and the eight term exact sequence In this section we will use the method developed in [12] to construct an eight term exact sequence in Leibniz homology. Lemma 5.4.1. Let 0ÑaÑgτ ÑhÑ0be a split extension of Leibniz algebras, i.e. there is a homomorphism of Leibniz algebras σ:hÑgsuch that τ˝σ“1h. Then the induced homomorphism of Leibniz algebras aNgÑgNg is injective. Proof. Denote the induced homomorphism of Leibniz algebras aNgÑgNg by α. We shall show that there exists a linear map gNgÑaNgwhich is aK-linear splitting for α. For each element xPgthere are unique aPaand hPhsuch that x“a`σphq. Let β:gbgÑaNgbe a linear map given by
126 5 A non-abelian exterior product of Leibniz algebras pa`σphqq b pa1`σph1qq β ÞÑ aNσph1q ` aNa1`σphq N a1for each a, a1Paand h, h1Ph. It is easy to check that βis a well-defined K-linear map and that βpIm dq “ 0, where d:gbgbgÑgbgis the boundary map in CL˚pgq. Thus, βinduces the linear map ¯ β:g‚gÑaNg. Now, let δ:g‚gÑgNgbe the linear map defined as in Section 5.3. Then, the linear map ¯ βδ´1:gNgÑaNg is such that ¯ βδ´1α“1aNg. Thus, αis injective. Theorem 5.4.2. Let 0ÑrÑfÑgÑ0be a free presentation of a Leibniz algebra g. Then there is an isomorphism HL3pgq – Ker ´θr,f:rNfÑr¯. Proof. According to Remark 5.2.4, for computing HL˚pgqwe can use an exact augmented simplicial Leibniz algebra ¨ ¨ ¨ f2 d2 0 Ñ Ñ Ñ d2 2 f1 d1 0 Ñ Ñ d1 1 f0 d0 0 Ñg such that fiis a free Leibniz algebra over a set, for each iě0,f0“fand Ker d0 0“r. Then, the long exact homotopy sequence derived from the following short exact sequence of simplicial Leibniz algebras 0Ñ rf˚,f˚s Ñ f˚ÑAbpf˚q Ñ 0, implies that HL3pgqis isomorphic to the first homotopy group of the following simplicial Leibniz algebra ¨ ¨ ¨ rf2,f2s d2 0 Ñ Ñ Ñ d2 2 rf1,f1s d1 0 Ñ Ñ d1 1 rf0,f0s. Hence, HL3pgq – Ker d1 0XKer d1 1X rf1,f1s{d2 2`Ker d2 0XKer d2 1X rf2,f2s˘. Since HL2pf0q “ 0and HL3pf1q “ 0, using Hopf formulas we have Ker d1 0X rf1,f1s “ rKer d1 0,f1s, Ker d2 0XKer d2 1X rf2,f2s “ rKer d2 0XKer d2 1,f2s`rKer d2 0,Ker d2 1s.
5.4 Third homology and the eight term exact sequence 127 Therefore, HL3pgq – Ker d1 1X rKer d1 0,f1s{d2 2`rKer d2 0XKer d2 1,f2s`rKer d2 0,Ker d2 1s˘ “Ker d1 1X rKer d1 0,f1s{`rd2 2pKer d2 0XKer d2 1q, d2 2pf2qs ` rd2 2pKer d2 0q, d2 2pKer d2 1qs˘ “Ker d1 1X rKer d1 0,f1s{`rKer d1 0XKer d1 1,f1s`rKer d1 0,Ker d1 1s˘. Since d1 1`rKer d1 0XKer d1 1,f1s`rKer d1 0,Ker d1 1s˘“0, we get HL3pgq – Ker ˜rKer d1 0,f1s rKer d1 0XKer d1 1,f1s`rKer d1 0,Ker d1 1s d1 1 ÝÑ rf0,f0s¸.(5.4.1) Furthermore, since 0ÑKer d1 1Ñf1 d1 1 Ñf0Ñ0is a free presentation of f0which splits, by Proposition 5.3.10 and Lemma 5.4.1 the following map Ker d1 1Nf1Ñ rKer d1 1,f1s, defined by xNyÞÑ rx, ys,yNxÞÑ ry, xs, for all xPKer d1 1and yPf1, is an isomorphism. Therefore, rKer d1 0,f1s rKer d1 0XKer d1 1,f1s`rKer d1 0,Ker d1 1s–Ker d1 0Nf1 `pKer d1 0XKer d1 1q N f1˘``Ker d1 0NKer d1 1˘. Hence, the exact sequences 0ÑKer d1 0XKer d1 1ÑKer d1 0ÑKer d0 0Ñ0and 0ÑKer d1 1Ñf1Ñf0Ñ0, and (5.3.1) imply that rKer d1 0,f1s rKer d1 0XKer d1 1,f1s`rKer d1 0,Ker d1 1s–Ker d0 0Nf0“rNf.(5.4.2) Now (5.4.1) and (5.4.2) complete the proof. Proposition 5.4.3. Let 0ÑaÑgτ ÑhÑ0be an extension of Leibniz algebras. Then we have the following exact sequence HL3pgq Ñ HL3phq Ñ Ker `θa,g:aNgÑa˘ÑHL2pgq Ñ HL2phq Ña{ra,gs Ñ HL1pgq Ñ HL1phq Ñ 0. Proof. Any free presentation 0ÑrÑfρ ÑgÑ0of gproduces a free presentation 0ÑsÑfτ˝ρ ÑhÑ0of hand an extension 0ÑrÑsÑaÑ0 of Leibniz algebras. By (5.3.1) we have the following exact sequence sNrˆrNfÑsNfÑaNgÑ0.
128 5 A non-abelian exterior product of Leibniz algebras This sequence yields the following exact sequence rNfÑsNfÑaNgÑ0. Thus, we have the following commutative diagram with exact rows rNf,2 θr,f sNf,2 θs,f aNg,2 θa,g 0 0,2r,2s,2a,20. The Snake Lemma and Theorem 5.4.2 imply the following exact sequence HL3pgq Ñ HL3phq Ñ Ker `θa,g:aNgÑa˘Ñr{rr,fs Ñ s{rs,fs. It is easy to see that Im `Ker `θa,g:aNgÑa˘Ñr{rr,fs˘ĎrX rf,fs rr,fs. Therefore, we have an exact sequence: HL3pgq Ñ HL3phq Ñ Ker `θa,g:aNgÑa˘ÑrX rf,fs rr,fsÑsX rf,fs rs,fs. Using the Hopf formula we get an exact sequence: HL3pgq Ñ HL3phq Ñ Ker `θa,g:aNgÑa˘ÑHL2pgq Ñ HL2phq. The rest of the proof follows from Proposition 5.3.13. Corollary 5.4.4. (see [9]) Let 0ÑaÑgÑhÑ0be a central extension of Leibniz algebras, i.e. ra, xs “ rx, as “ 0for all aPaand xPg. Then there is the following exact sequence HL3pgq Ñ HL3phq Ñ Cokerˆabaη Ñabg rg,gs‘g rg,gsba˙ ÑHL2pgq Ñ HL2phq Ñ aÑHL1pgq Ñ HL1phq Ñ 0, where η:abaÑabg rg,gs‘g rg,gsbais given by abbÞÑ pabb, ´abbq, where a“a` rg,gsand b“b` rg,gsfor each a, b Pa.
5.5 Relationship to the universal quadratic functor 129 Proof. Under the required conditions, the Leibniz algebras aand gact trivially on each other. Then, by [18, Proposition 4.2], we have a natural isomorphism a‹g–abg rg,gs‘g rg,gsba. Since aNgis obtained from a‹gby killing the elements of the form a˚ ipbq ´ ipaq ˚ b, where a, b Paand i:aÑgis the natural inclusion, we get an isomorphism aNg–Coker´η:abaÑabg rg,gs‘g rg,gsba¯. Then, by Proposition 5.4.3 we conclude the required result. 5.5 Relationship to the universal quadratic functor In this section Kis a commutative ring with identity (not necessarily a field). Keeping in mind Remark 5.3.6, we will use only those constructions and facts from the previous sections, which do not require Kto be a field. In the case of Lie algebras, there is a connection between the non-abelian exterior product of Lie algebras and Whitehead’s universal quadratic functor ([15]). We can observe it in the Leibniz algebras case too. Definition 5.5.1 ([29]).Let Abe a K-module and consider the endofunctor that sends Ato the K-module generated by the symbols γpaqwith aPA, quotient by the submodule generated by k2γpaq “ γpkaq, γpa`b`cq ` γpaq ` γpbq ` γpcq “ γpa`bq ` γpa`cq ` γpb`cq, γpka `bq ` kγpaq ` kγpbq “ kγpa`bq ` γpkaq ` γpbq, for all kPKand a, b, c PA. This functor denoted by Γis called universal quadratic functor. Proposition 5.5.2 ([29]).Let Ibe a well-ordered set and Abe a free K-module with basis teiuiPI. Then ΓpAqis a free K-module with basis tγpeiquiPIY tγpei`ejq ´ γpeiq ´ γpejquiăj
130 5 A non-abelian exterior product of Leibniz algebras Let η:mÑgand µ:nÑgbe two crossed modules of Leibniz algebras. As we know there are induced actions of mand non each other via the action of g. Let mˆgn“ tpm, nq | ηpmq “ µpnqu be the pullback of ηand µ. It is a Leibniz subalgebra of m‘n. Let xm,ny “ tpτmpxq, τnpxqq | xPm‹nu, where τm:m‹nÑmand τn:m‹nÑnare homomorphisms introduced in Subsection 5.3.2. Proposition 5.5.3. xm,nyis an ideal of mˆgnand the quotient pmˆgnq{xm,ny is abelian. Proof. The assertion that xm,nyis an ideal of mˆgnfollows by straightforward calculations. For instance, given any mPm,nPnand pm1, n1q P mˆgnwe get rpτmpm˚nq, τnpm˚nqq,pm1, n1qs “ rpmn,mnq,pm1, n1qs “ prmn, m1s,rmn, n1sq “ ppmnqηpm1q,µpmnqn1q “ ppmnqn1,pmnqn1q “ pτmpmn˚n1q, τnpmn˚n1qq. Now take any pm, nq,pm1, n1q P mˆgn, then we have rpm, nq,pm1, n1qs “ prm, m1s,rn, n1sq “ pmm1,nn1q “ pmn1,mn1q “ pτmpm˚n1q, τnpm˚n1qq, showing that pmˆgnq{xm,nyis abelian. In the subsequent statements, given a K-module A, we consider the Kmodule ΓpAqas an abelian Leibniz algebra. Proposition 5.5.4. There is a well-defined homomorphism of Leibniz algebras Γˆmˆgn xm,ny˙ψ,2m‹n, given by ψpγppm, nq ` xm,nyqq “ m˚n´n˚m.
5.5 Relationship to the universal quadratic functor 131 Proof. It is easy to check that ψpreserves the defining relations of Γ. Thus, it suffices to show that pm1`τmpxqq ˚ pn1`τnpxqq ´ pn1`τnpxqq ˚ pm1` τmpxqq “ m1˚n1´n1˚m1for each xPm‹n. This reduces to prove that m1˚mn´mn˚m1`mn˚n1´n1˚mn“0. Using the defining relations of the non-abelian tensor product we have m1˚mn´mn˚m1`mn˚n1´n1˚mn “rm1, ms ˚ n´m1n˚m´m˚nm1´ rm, m1s ˚ n `mn1˚n`m˚ rn, n1s ´ n1m˚n` rn1, ns ˚ m “rm1, ms ˚ n´ rn1, ns ˚ m´m˚ rn, n1s´rm, m1s ˚ n ` rm, m1s ˚ n`m˚ rn, n1s`rm1, ms ˚ n´ rn1, ns ˚ m“0. By Proposition 5.3.4 we know that Im ψis contained in the centre of m‹n, so ψis a homomorphism of Leibniz algebras. It is clear that Im ψis contained in Kerpπ:m‹nÑmNnq, where πis the canonical projection. But the following sequence Γˆmˆgn xm,ny˙ψ,2m‹nπ,2mNn,20, is not exact in many cases. Nevertheless, we have the following Proposition 5.5.5. There is an exact sequence of Leibniz algebras Γˆmˆgn xm,ny‘mˆgn xm,ny˙r ψ,2m‹nπ,2mNn,20, where r ψpγppm, nq ` xm,ny,pm1, n1q`xm,nyqq “ m˚n1´n˚m1. Proof. Like in Proposition 5.5.4, the crucial part of the proof is to show that pm`τmpxqq ˚ pn1`τnpx1qq ´ pn`τnpxqq ˚ pm1`τmpx1qq “ m˚n1´n˚m1 for all x, x1Pm‹n. Let x“m1˚n1and x1“m1 1˚n1 1, then proving that m˚m1 1n1 1`mn1 1˚n1´n˚m1n1 1 1´m1n1˚m1“0, will imply the result. Using the defining identities of the non-abelian tensor product, we get m˚m1 1n1 1`mn1 1˚n1´n˚m1n1 1 1´m1n1˚m1 “rm, m1 1s ˚ n1 1´mn1 1˚m1 1`m1˚ rn1, n1s ` mn1 1˚n1 ` rn, n1 1s ˚ m1 1´nm1 1˚n1 1´ rm1, m1s ˚ n1´m1˚nm1 1“0.
234 11 A characterisation of Lie algebras x¨yb2is zero in the sum B5X`B5Y. Recall that b2RB5Y, so that yb2cannot be decomposed as a product of yand b2in B5Y. By Theorem 11.2.1, yb2can also not be written as a product in which more than one yor b2appears, unless yb2is zero in B5Y. As a consequence, x¨yb2can only be zero if either yb2is zero in B5Y, or xz “0is a law in V. In the former case, yb2is zero in the sum B`Y, which is a free algebra on tb2, yu; then yz “0is a law in V. In either case, Vis abelian. 11.2.6 (Anti)associative algebras Essentially the same argument gives us two further examples, which we shall need later on: Proposition 11.2.3. If a variety of either associative or antiassociative algebras is locally algebraically cartesian closed, then it is abelian. Proof. In the antiassociative case we have x,´xb1PB5Xand b2y,yPB5Y such that x¨b2yand ´xb1¨yare sent to the same element in B5pX`Yqby the above isomorphism pB5ιXB5ιYq:B5X`B5YÑB5pX`Yq. Similarly, in the associative case, we see that xb1¨yand x¨b2yare two distinct elements of the sum B5X`B5Ywhich the morphism pB5ιXB5ιYq sends to one and the same element of B5pX`Yq. Lemma 11.2.4. Any variety of K-algebras that satisfies the law xpxyq “ 0is a subvariety of AAAlgK. Proof. Taking x“a`band y“cgives us 0“ pa`bqppa`bqcq “ apacq ` bpacq ` apbcq ` bpbcq “ bpacq ` apbcq so that apbcq “ ´bpacq. It follows that puvqw“ ´wpuvq “ upwvq “ ´upvwq is a law in V, and Vis a variety of antiassociative algebras. 11.2.7 Algebraic coherence Theorem 11.2.1 gives us a characterisation of algebraic coherence for varieties of K-algebras. Theorem 11.2.5. Let Kbe an infinite field. If Vis a variety of nonassociative K-algebras, then the following are equivalent:
11.2 Main result 235 (i) Vis algebraically coherent; (ii) there exist λ1, ..., λ16 in Ksuch that zpxyq “ λ1ypzxq ` λ2xpyzq ` λ3ypxzq ` λ4xpzyq `λ5pzxqy`λ6pyzqx`λ7pxzqy`λ8pzyqx pxyqz“λ9ypzxq ` λ10xpyzq ` λ11ypxzq ` λ12xpzyq `λ13pzxqy`λ14pyzqx`λ15pxzqy`λ16pzyqx are laws in V; (iii) Vis an Orzech category of interest [28]. Proof. From the results of [10] we already know that (iii) implies (i). It follows immediately from the definition of an Orzech category of interest that (ii) implies (iii). To see that (i) implies (ii), we take free B-actions as in the first part of the proof of Proposition 11.2.2 and obtain the regular epimorphism pB5ιXB5ιYq:B5X`B5YÑB5pX`Yq. Any element bpxyqof B5pX`Yqis the image through this morphism of some polynomial ψpb1, x, b2, yqin B5X`B5Y. Note that this polynomial cannot contain any monomials obtained as a product of a biwith xy or yx. This allows us to write, in the sum B`X`Y, the element bpxyqas λ1ypbxq ` λ2xpybq ` λ3ypxbq ` λ4xpbyq ` λ5pbxqy`λ6pybqx `λ7pxbqy`λ8pbyqx`νφpb, x, yq for some λ1, ..., λ8,νPK, where φpb, x, yqis the part of the polynomial in b, xand ywhich is not in the homogeneous component of bpxyq. Since B`X`Y is the free V-algebra on three generators b,xand y, from Theorem 11.2.1 we deduce that the first equation in (ii) is again a law in V. Analogously for pxyqb we deduce the second equation in (ii). Remark 11.2.6.This result may be used to prove the claim made in [10] that the category of Jordan algebras—commutative and such that pxyqpxxq “ xpypxxqq—is not algebraically coherent. Indeed, as explained in [27], it is not acategory of interest.
236 11 A characterisation of Lie algebras In the case of alternating algebras, this characterisation becomes more precise: Theorem 11.2.7. Let Kbe an infinite field. If Vis a subvariety of AltK, then the following are equivalent: (i) Vis algebraically coherent; (ii) Vis a subvariety of either AAAlgKor LieK. Proof. (ii) implies (i) since AAAAlgKand LieKare Orzech categories of interest [28], so their subvarieties are algebraically coherent. To see that (i) implies (ii), we first use anticommutativity to simplify the law given in Theorem 11.2.5 to zpxyq “ λypzxq ` µxpyzq for some λand µin K. Choosing, in turn, y“zand x“z, we see that 1. either λ“ ´1or z¨zx “0is a law in V, and 2. either µ“ ´1or x¨xy “0is a law in V. In any of the latter cases, Vis a variety of antiassociative algebras by Lemma 11.2.4, which makes it abelian (Proposition 11.2.3). We are left with the situation when λ“µ“ ´1, which means that the Jacobi identity is a law in V, so that Va variety of Lie algebras. Example 11.2.8. The variety of alternating associative algebras is an example. We have that 0“xpyyq “ pxyqyis a law, so that by Lemma 11.2.4 those algebras are antiassociative as well. It follows that xyz “0is a law. We regain a variety as in Proposition 11.2.2, so since it is not abelian, it cannot be (LACC). 11.2.8 A characterisation of Lie algebras amongst alternating algebras The condition (LACC) eliminates one of the two options in Theorem 11.2.7. Theorem 11.2.9. Let Kbe an infinite field. If Vis a locally algebraically cartesian closed variety of alternating K-algebras, then it is a subvariety of LieK. In other words, LieKis the largest (LACC) variety of alternating K-algebras. Thus for any variety Vof alternating K-algebras, the following are equivalent:
11.3 Non-alternating algebras 237 (i) Vis a subvariety of a (LACC) variety of alternating K-algebras; (ii) the Jacobi identity is a law in V. Proof. This combines Theorem 11.2.7 with Proposition 11.2.3. Remark 11.2.10.By Proposition 11.2.2, the condition (iii) Vis (LACC) is strictly stronger than the equivalent conditions (i) and (ii). Remark 11.2.11.We do not know any non-abelian examples of (LACC) subvarieties of LieK, or whether such subvarieties even exist. 11.3 Non-alternating algebras An important question which we have to leave open for now, is what happens when the algebras we consider are not alternating. We end this note with some of our preliminary findings. Some of the results and techniques used in the previous section are valid for non-alternating algebras of course. For instance, Proposition 11.2.2, Proposition 11.2.3 and Theorem 11.2.5 are. Proposition 11.2.2 contradicts—in spirit only, since we are working over a field—Proposition 6.9 in [17], which claims that the category of all commutative non-unitary rings satisfying xyz “0is locally algebraically cartesian closed. It turns out that the argument given in Proposition 11.2.2 is still valid in this case, and shows that the variety under consideration, should it be (LACC), would be abelian—which is false. We noticed that the functor Rconstructed in the proof of [17, Proposition 6.9] is not well defined on morphisms. Let us give a concrete example showing this in detail. We follow the notations from [17, Proposition 6.9]. Let B“ xby “ Zrbsact on the commutative ring X“ xx, y |xx “xy “yy “0y “ Zrx, ys{pxx, xy, yyq by bx “yand by “0. Consider M“ xm, p, q |mp “q, mq “pq “mm “pp “qq “0y
238 11 A characterisation of Lie algebras with the trivial B-action, and let g:XÑMbe the ring homomorphism sending xand yto m. Let fPRpXqbe defined by fpn, bq “ nx `bx. Then Rpgqpfqp0, bq ¨ p“ pg˝fqp0, bq ¨ p“gpyq ¨ n“mp “q‰0, which shows that Rpgqpfqis not an element of RpMq. 11.3.1 Leibniz algebras The category LeibKof (right) Leibniz algebras [23] over Kis the subvariety of AlgKsatisfying the (right) Leibniz identity pxyqz“xpyzq ` pxzqy. This law is clearly equivalent to the Jacobi identity when the algebras are alternating, so that a Lie algebra is the same thing as an alternating Leibniz algebra. However, examples of non-alternating Leibniz algebras exist. Analogously, we can consider the category of (left) Leibniz algebras, with corresponding identity xpyzq“pxyqz`ypxzq. Both categories are of course equivalent. We do not know whether Theorem 11.2.9 extends to the non-alternating case. What is certain, though, is that the category of Leibniz algebras is not locally algebraically cartesian closed. Indeed, using the notations of Proposition 11.2.2, the Leibniz identity allows us to deduce #pxyqb“xpybq`pxbqy pxyqb“ ´xpbyq`pxbqy so that x¨yb “ ´x¨by in B`X`Y. This means that x¨yb2and ´x¨b2y are two distinct elements of B5X`B5Ywhich are sent to the same element of B5pX`Yqby the morphism pB5ιXB5ιYq. Hence this morphism cannot be an isomorphism, and LeibKis not (LACC). We may ask ourselves what happens in the “intersection” between right and left Leibniz algebras. They are called symmetric Leibniz algebras and, as shown in [26], the chain of inclusions LieKĎSLeibKĎLeibKis strict. Doing a rearrangement of terms as in bpxyq “ pbxqy`xpbyq “ bpxyq`pbyqx`xpbyq, we see that pbyqx`xpbyq “ 0. From this we may conclude that, in order to be (LACC), a variety of symmetric Leibniz algebras must either be alternating or abelian. We thus regain the known cases of Lie algebras and vector spaces.
11.3 Bibliography 239 11.3.2 Free algebra on one generator A variety of algebras is alternating precisely when the free algebra on a single generator is abelian: xx “0is a law in the variety, if and only if the bracket vanishes on the algebra freely generated by txu. This corresponds to the condition that the free algebra on a single generator admits an internal abelian group structure. This condition makes sense in arbitrary semi-abelian varieties, and we may ask ourselves whether perhaps it is implied by (LACC), as in the case of symmetric Leibniz algebras. This would allow us to drop the condition that Vis alternating in Theorem 11.2.9. The example of crossed modules proves that this is false. In [9] it is shown that on the one hand, a crossed module B:TÑGwith action ξadmits an internal abelian group structure if and only if the groups Tand Gare abelian and the action ξis trivial. On the other hand, the free crossed module on a single generator is the inclusion κZ,Z:Z5ZÑZ`Z, equipped with the conjugation action. We see that in this case the free object on one generator is not abelian, even though XMod is a locally algebraically cartesian closed semi-abelian variety. However, it is not a variety of non-associative algebras of course. Perhaps this is not the right conceptualisation, and we must think of other ways of making the law xx “0categorical. The question then becomes whether (LACC), or any other appropriate categorical-algebraic condition, would imply this new characterisation. Acknowledgements This work was partially supported by Ministerio de Economía y Competitividad (Spain), grant MTM2016-79661-P. The first author was also supported by Xunta de Galicia, grant GRC2013-045 (European FEDER support included), by an FPU scholarship of the Ministerio de Educación, Cultura y Deporte (Spain) and by a Fundación Barrié scholarship. The second author is a Research Associate of the Fonds de la Recherche Scientifique–FNRS Thanks to James R. A. Gray, George Janelidze and Zurab Janelidze for fruitful discussions and important comments on our work. We would also like to thank the University of Cape Town and Stellenbosch University for their kind hospitality during our stay in South Africa.
240 11 A characterisation of Lie algebras Bibliography [1] M. Barr, Coalgebras over a commutative ring, J. Algebra 32 (1974), 600– 610. [2] F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semiabelian categories, Math. Appl., vol. 566, Kluwer Acad. Publ., 2004. [3] D. Bourn, Normalization equivalence, kernel equivalence and affine categories, Category theory (Como, 1990), Lecture Notes in Math., vol. 1488, Springer, Berlin, 1991, pp. 43–62. [4] D. Bourn, Normal functors and strong protomodularity, Theory Appl. Categ. 7(2000), no. 9, 206–218. [5] D. Bourn, 3ˆ3Lemma and protomodularity, J. Algebra 236 (2001), 778–795. [6] D. Bourn, The cohomological comparison arising from the associated abelian object, preprint arXiv:1001.0905, 2010. [7] D. Bourn and J. R. A. Gray, Aspects of algebraic exponentiation, Bull. Belg. Math. Soc. Simon Stevin 19 (2012), 823–846. [8] D. Bourn and G. Janelidze, Protomodularity, descent, and semidirect products, Theory Appl. Categ. 4(1998), no. 2, 37–46. [9] P. Carrasco, A. M. Cegarra, and A. R.-Grandjeán, (Co)Homology of crossed modules, J. Pure Appl. Algebra 168 (2002), no. 2-3, 147–176. [10] A. S. Cigoli, J. R. A. Gray, and T. Van der Linden, Algebraically coherent categories, Theory Appl. Categ. 30 (2015), no. 54, 1864–1905. [11] A. S. Cigoli, J. R. A. Gray, and T. Van der Linden, On the normality of Higgins commutators, J. Pure Appl. Algebra 219 (2015), 897–912. [12] F. Conduché, Au sujet de l’existence d’adjoints à droite aux foncteurs “image réciproque” dans la catégorie des catégories, C. R. Acad. Sci. Paris Sér. A-B 275 (1972), A891–A894.
11.3 Bibliography 241 [13] M. Gerstenhaber, A categorical setting for the Baer extension theory, Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVII, New York, 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 50–64. [14] M. Gran, G. Kadjo, and J. Vercruysse, A torsion theory in the category of cocommutative Hopf algebras, Appl. Categ. Structures 24 (2016), no. 3, 269–282. [15] J. R. A. Gray, Algebraic exponentiation in general categories, Ph.D. thesis, University of Cape Town, 2010. [16] J. R. A. Gray, Algebraic exponentiation for categories of Lie algebras, J. Pure Appl. Algebra 216 (2012), 1964–1967. [17] J. R. A. Gray, Algebraic exponentiation in general categories, Appl. Categ. Structures 20 (2012), 543–567. [18] P. J. Higgins, Groups with multiple operators, Proc. London Math. Soc. (3) 6(1956), 366–416. [19] S. A. Huq, Commutator, nilpotency, and solvability in categories, Quart. J. Math. Oxford Ser. (2) 19 (1968), 363–389. [20] G. Janelidze, L. Márki, and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002), no. 2-3, 367–386, Category theory 1999 (Coimbra). [21] P. T. Johnstone, Sketches of an elephant: A topos theory compendium, Oxford Logic Guides, vol. 43, 44, Oxford Univ. Press, 2002, Volumes 1 and 2. [22] R. Lavendhomme and J. R. Roisin, Cohomologie non abélienne de structures algébriques, J. Algebra 67 (1980), 385–414. [23] J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. (2) 39 (1993), no. 3-4, 269–293. [24] S. Mac Lane, Categories for the working mathematician, second ed., Grad. Texts in Math., vol. 5, Springer, 1998. [25] N. Martins-Ferreira and T. Van der Linden, A note on the “Smith is Huq” condition, Appl. Categ. Structures 20 (2012), no. 2, 175–187.
242 11 A characterisation of Lie algebras [26] G. Mason and G. Yamskulna, Leibniz algebras and Lie algebras, SIGMA Symmetry Integrability Geom. Methods Appl. 9(2013), Paper 063, 10. [27] A. Montoli, Action accessibility for categories of interest, Theory Appl. Categ. 23 (2010), no. 1, 7–21. [28] G. Orzech, Obstruction theory in algebraic categories. I, II, J. Pure Appl. Algebra 2(1972), 287–314; ibid. 2 (1972), 315–340. [29] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov, Rings that are nearly associative, Academic Press, 1982.
Chapter 12 Future research At the moment of writing this dissertation, several ongoing joint projects are being worked on. One such with José Manuel Casas and Tim Van der Linden is to study low-dimensional homology of multiplicative Hom-Lie algebras [7]. This category is of special interest since, even though it has some good algebraic behaviour, it is an example of semi-abelian category where the Smith is Huq condition [10], normality of Higgins commutators [2] or even the universal central extension condition [1] do not hold. A joint project with Rafael Fernández-Casado is to generalise the notion of actor of crossed modules of Leibniz algebras to the dialgebras setting [8]. The aim is to extend Chapter 6 to complete the square-shaped diagram formed by associative algebras, Lie algebras, dialgebras and Leibniz algebras: XAs XLieAs ,2 i K % XLie i XU lr $ XDial XLb ,2 XAs LR JXLb XUd lr XLieLb LR Note that the first two cases were studied in my coauthor’s Ph.D. thesis [4]. An other ongoing project with James Gray is to prove that all categories of Lie objects over an abelian, cocomplete, symmetric, closed, monoidal category are (LACC) [5], generalizing his proof in the case of R-modules [6]. This 243
A rA, AsbKHC1pAqH1`A, VpAq˘H1pA, rA, Asq HC1pAqHCM 1pAqrA, As “A, rA, As‰0. Teorema 2.6.6. Sexa Mun ideal graduado dunha superálxebra de Lie P. Entón existe una sucesión exacta KerpP^MÑPq Ñ H2pPq Ñ H2pP{Mq Ñ M rP, MsÑH1pPq Ñ H1pP{Mq Ñ 0. No Capítulo 3 atópase a extensión central universal das superálxebras de matrices slpm, n, Aqonde Aé unha superálxebra asociativa e m`n“3,4, relacionándoa coa superálxebra de Steinberg stpm, n, Aq. Calcúlanse H2`slpm, n, Aq˘eH2`stpm, n, Aq˘, e para rematar, introdúcese un novo método usando o produto tensor non abeliano de superálxebras de Lie (definido no capítulo anterior) para atopar unha conexión entre H2`slpm, n, Aq˘e a homoloxía cíclica de superálxebras asociativas cando m`ně3. Teorema 3.8.1. Sexa Kun anel conmutativo con unidade e Aunha Ksuperálxebra asociativa con unidade. Entón, H2`stpm, n, Aq˘“ $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % 0for m`ně5or m“2, n “1, A6 3for m“3, n “0, A6 2for m“4, n “0, ΠpA2q6for m“3, n “1, A4 2‘A2 0for m“2, n “2, onde Amé o cociente de Apolo ideal mA `ArA, As(Definición 3.2.3) e Πé o funtor cambio de paridade. Teorema 3.8.2. Sexa Kun anel conmutativo con unidade e Aunha K-superálxebra asociativa con unidade con unha K-base que conteña a 250
unidade. Entón, H2`slpm, n, Aq˘“ $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % HC1pAqfor m`ně5or m“2, n “1, HC1pAq ‘ A6 3for m“3, n “0, HC1pAq ‘ A6 2for m“4, n “0, HC1pAq ‘ ΠpA2q6for m“3, n “1, HC1pAq ‘ A4 2‘A2 0for m“2, n “2, onde Amé o cociente de Apolo ideal mA `ArA, As(Definición 3.2.3) e Πé o funtor cambio de paridade. Seguindo as liñas do capítulo anterior, no Capítulo 4 complétase o problema de atopar a extensión central universal na categoría de superálxebras de Leibniz das superálxebras de matrices slpm, n, Dqcando m`ně3eD é unha superdiálxebra, resolvendo o problema particular de cando se trata dunha álxebra asociativa, superálxebra ou diálxebra. Para completar esta tarefa, empregamos un método orixinal distinto do habitual que se pode atopar na literatura. Tamén se introduce o cadrado tensor non abeliano de álxebras de Leibniz para estudar as súas relacións coa extensión central universal. Teorema 4.6.1. Sexa Run anel conmutativo con unidade e Dunha Rsuperdiálxebra asociativa con unidade e unha R-base que conteña a barunidade. Entón, HL2`slpm, n, Dq˘“ $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % HHS1pDqfor m`ně5or m“2, n “1, HHS1pDq ‘ D6 3for m“3, n “0, HHS1pDq ‘ D6 2for m“4, n “0, HHS1pDq ‘ ΠpD2q6for m“3, n “1, HHS1pDq ‘ D4 2‘D2 0for m“2, n “2, onde Dme o resultado de cocientar Dpolo ideal mD ` prD, Ds % Dq(Definición 4.4.2) e Πé o funtor cambio de paridade. Teorema 4.6.2. Sexa Run anel conmutativo con unidade e Dunha Rsuperdiálxebra asociativa con unidade e unha R-base que conteña a bar251
unidade. Entón, HL2`stlpm, n, Dq˘“ $ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % 0for m`ně5or m“2, n “1, D6 3for m“3, n “0, D6 2for m“4, n “0, ΠpD2q6for m“3, n “1, D4 2‘D2 0for m“2, n “2, onde Dme o resultado de cocientar Dpolo ideal mD ` prD, Ds % Dq(Definición 4.4.2) e Πé o funtor cambio de paridade. No Capítulo 5 introdúcese o produto exterior non abeliano de dous módulos cruzados de álxebras de Leibniz e investígase a súa relación coa homoloxía de Leibniz en dimensión baixa. Este produto aplícase á construción dunha sucesión exacta de oito termos. Por último, establécese unha relación co funtor cuadrático universal, aplicándoo a unha comparación entre a segunda homoloxía de Lie e a segunda homoloxía de Leibniz. Corolario 5.4.4. Sexa 0ÑaÑgÑhÑ0unha extensión central de álxebras de Leibniz, é dicir, ra, xs “ rx, as “ 0para todo aPaexPg. Entón existe a seguinte sucesión exacta: HL3pgq Ñ HL3phq Ñ Cokerˆabaη Ñabg rg,gs‘g rg,gsba˙ ÑHL2pgq Ñ HL2phq Ñ aÑHL1pgq Ñ HL1phq Ñ 0, onde η:abaÑabg rg,gs‘g rg,gsbaven dado por abbÞÑ pabb, ´abbq, con a“a` rg,gseb“b` rg,gspara cada a, b Pa. Proposición 5.6.1. Sexa gunha álxebra de Lie. Entón, existe un subespazo vectorial Vde Kerttg:HL2pgq Ñ H2pgqu tal que temos un epimorfismo VÑΓpgabq. De este xeito, se gnon é unha álxebra de Lie perfecta, tg:HL2pgq Ñ H2pgqnon é un isomorfismo. No Capítulo 6 esténdese a noción de biderivación á categoría de módulos cruzados de álxebras de Leibniz a través das accións de álxebras de Leibniz. Isto permítenos construír un obxecto que é o actor baixo certas circunstancias. 252
Ademais, dáse unha descrición da acción na categoría de módulos cruzados de álxebras de Leibniz en termos de ecuacións. Para rematar, compróbase que baixo certas condicións, o núcleo do morfismo canónico que vai dende un módulo cruzado ao seu actor coincide co seu centro, e introducimos as nocións de biderivacións internas e externas en módulos cruzados. Teorema 6.4.3. Sexa pm,p, ηqepn,q, µqen XLb. Sempre que se cumpran certas condicións de compatibilidade, existe un homomorfismo de módulos cruzados dende pm,p, ηqapBiderpq,nq,Biderpn,q, µq,∆q. Ademais, o recíproco tamén é certo se estamos nalgún dos seguintes casos: Annpnq “ 0“Annpqq,(CON1) Annpnq “ 0erq,qs “ q,(CON2) rn,ns “ nerq,qs “ q.(CON3) No Capítulo 7 esténdese a módulos cruzados o funtor álxebra envolvente universal entre álxebras asociativas e álxebras de Leibniz. Constrúese un isomorfismo entre a categoría de representacións dun módulo cruzado de álxebras de Leibniz e a categoría de módulos pola esquerda sobre o seu módulo cruzado envolvente universal. O modo de estudar o problema é especialmente interesante xa que o actor na categoría de módulos cruzados de álxebras de Leibniz non existe por norma xeral, así que a proba no caso de Lie non pode ser aplicada. Para rematar, estudamos o devandito funtor dentro do marco da categoría de Loday-Pirashvili [31] para entender este módulo cruzado envolvente universal en termos do caso de módulos cruzados de álxebras de Lie. Teorema 7.5.1. A categoría de representacións dun módulo cruzado de álxebras de Leibniz pq,p, ηqé isomorfa á categoría de módulos pola esquerda sobre o seu módulo cruzado de álxebras envolvente universal XULpq,p, ηq. No Capítulo 8 investígase en que sentido, para ně3, as n-álxebras de Lie admiten álxebras envolventes universais. Houbo varios intentos dunha construción (ver [14] e [2]) mais despois de analizalas polo miúdo chegamos á conclusión de que en xeral non son válidas. Para isto, danse contraexemplos e condicións suficientes. Logo, analizamos o problema en toda a súa xeneralidade, demostrando que a universalidade é incompatible co feito de que a categoría de módulos sobre unha n-álxebra de Lie dada sexa equivalente á categoría de módulos sobre a álxebra asociada UpLq. De feito, existe cando 253
menos un funtor álxebra asociada U: n-LieKÑAlgKque induce esta equivalencia, pero nunca admite un adxunto pola dereita. Para rematar, defínese una teoría de (co)homoloxía baseada no funtor álxebra asociada U. Proposición 8.3.9. BLbΛ n´1pLq – InnDerpLqse e só se Kn´1“Wn´1. Teorema 8.4.6. O funtor U: n-LieKÑAlgKten un adxunto pola dereita se e só se n“2. Máis concretamente, para ną2non existe ningún funtor F:n-LieKÑAlgKcun adxunto pola dereita G:AlgKÑn-LieKque induza unha equivalencia de categorías entre L-ModKeModFpLqpara todo L. No Capítulo 9 próbase que un monoide Mé un grupo se e só se, na categoría de monoides, todos os puntos sobre Mson fortes. Este resultado mellora e simplifica o traballo e Montoli, Rodelo e Van der Linden [32] que caracteriza os grupos entre os monoides coma os obxectos protomodulares. Teorema 9.0.1. Un monoide Mé un grupo se e só se, en Mon, todos os puntos sobre Mson fortes. No Capítulo 10 dáse unha caracterización universal das álxebras de Hopf entre as biálxebras coconmutativas sobre un corpo alxebricamente pechado: una biálxebra coconmutativa é unha álxebra de Hopf cando toda extensión escindida sobre ela admite unha descomposición. Tamén se resolve que este resultado non pode ser estendido ao contexto non coconmutativo, probando así que as categorías de biálxebras e álxebras de Hopf non son unitais nin protomodulares. Teorema 10.3.5. Se Ké un corpo alxebricamente pechado e Yé unha biálxebra coconmutativa sobre K, entón as seguintes condicións son equivalentes: (i) Yé unha álxebra de Hopf; (ii) en BiAlgK,coc, todas as extensións escindidas sobre Yadmiten unha descomposición; (iii) Yé un obxecto protomodular en BiAlgK,coc . Proposición 10.4.1. Se Yé un obxecto unital de BiAlgK, entón para todo X temos un isomorfismo XˆY–XbY. 254
No Capítulo 11 próbase que se Ké un corpo infinito, unha variedade de K-álxebras alternadas—non necesariamente asociativas, onde se cumpre xx “0—é localmente alxebricamente cartesiana pechada (segundo a definición de Gray [25]), entón é unha variedade de álxebras de Lie. En particular, LieKé a variedade máis grande. Deste xeito, para unha variedade de K-álxebras alternadas, a identidade de Jacobi convértese nunha condición categórica. Tamén se dá unha caracterización das variedades coherentes alxebricamente (segundo Cigoli, Gray e Van der Linden [12]). Teorema 11.2.9. Sexa Kun corpo infinito. Se Vé unha localmente alxebricamente cartesiana pechada de K-álxebras alternadas, entón é unha subvariedade de LieK. En outras palabras, LieKé a maior (LACC) variedade de K-álxebras alternadas. Entón, para unha variedade de K-álxebras alternadas, son equivalentes: (i) Vé unha subvariedade dunha variedade (LACC) K-álxebras alternadas; (ii) a identidade de Jacobi cúmprese en V. Teorema 11.2.5. Sexa Kun corpo infinito. Se Vé una variedade de Kálxebras non asociativas, entón son equivalentes: (i) Vé coherente alxebricamente; (ii) existen λ1, ..., λ16 in Ktales que as identidades zpxyq “ λ1ypzxq ` λ2xpyzq ` λ3ypxzq ` λ4xpzyq `λ5pzxqy`λ6pyzqx`λ7pxzqy`λ8pzyqx pxyqz“λ9ypzxq ` λ10xpyzq ` λ11ypxzq ` λ12xpzyq `λ13pzxqy`λ14pyzqx`λ15pxzqy`λ16pzyqx cúmprense en V; (iii) Vé unha categoría de interese de Orzech [34]. Por último, no Capítulo 12 fálase sobre diversos traballos de investigación que o autor está a realizar neste momento, e sobre ideas que poden xurdir tendo esta tese como punto de partida. 255
Bibliografía [1] J. Arnlind, A. Kitouni, A. Makhlouf, and S. Silvestrov, Structure and cohomology of 3-Lie algebras induced by Lie algebras, Geometry and Mathematical Physics (AGMP, Mulhouse, France), Springer Proceedings in Mathematics & Statistics, vol. 85, 2014, pp. 123–144. [2] D. Bălibanu and J. van de Leur, Irreducible highest weight representations of the simple n-Lie algebra, Transform. Groups 17 (2012), no. 3, 593–613. [3] M. Barr, Exact categories, Exact Categories and Categories of Sheaves, Lecture Notes in Math., vol. 236, Springer, Berlin, 1971, pp. 1–120. [4] F. Borceux and D. Bourn, Mal’cev, protomodular, homological and semiabelian categories, Math. Appl., vol. 566, Kluwer Acad. Publ., 2004. [5] D. Bourn, Normalization equivalence, kernel equivalence and affine categories, Category theory (Como, 1990), Lecture Notes in Math., vol. 1488, Springer, Berlin, 1991, pp. 43–62. [6] J. M. Casas, R. Fernández-Casado, X. García-Martínez, and E. Khmaladze, Actor of a crossed module of Leibniz algebras, preprint arXiv:1606.04871, 2016. [7] J. M. Casas, N. Inassaridze, and M. Ladra, Homological aspects of Lie algebra crossed modules, Manuscripta Math. 131 (2010), no. 3-4, 385– 401. [8] J. M. Casas and M. Ladra, Homology of crossed modules in Lie algebras, Bull. Soc. Math. Belg. Sér. A 45 (1993), no. 1-2, 59–84, Third Week on Algebra and Algebraic Geometry (SAGA III) (Puerto de la Cruz, 1992). [9] J. M. Casas, J.-L. Loday, and T. Pirashvili, Leibniz n-algebras, Forum Math. 14 (2002), no. 2, 189–207. [10] J. L. Castiglioni, X. García-Martínez, and M. Ladra, Universal central extensions of Lie-Rinehart algebras, J. Algebra Appl., 2017, doi:10.1142/S0219498818501347. [11] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. 256
[12] A. S. Cigoli, J. R. A. Gray, and T. Van der Linden, Algebraically coherent categories, Theory Appl. Categ. 30 (2015), no. 54, 1864–1905. [13] G. Donadze, X. García-Martínez, and E. Khmaladze, A non-abelian exterior product and homology of Leibniz algebras, Rev. Mat. Complut., 2017, doi:10.1007/s13163-017-0237-2. [14] A. S. Dzhumadil’daev, Representations of vector product n-Lie algebras, Comm. Algebra 32 (2004), no. 9, 3315–3326. [15] G. J. Ellis, A nonabelian tensor product of Lie algebras, Glasgow Math. J. 33 (1991), no. 1, 101–120. [16] R. Fernández-Casado, X. García-Martínez, and M. Ladra, A natural extension of the universal enveloping algebra functor to crossed modules of Leibniz algebras, Appl. Categ. Structures, 2016, doi:10.1007/s10485-0169472-9. [17] R. Fernández-Casado, X. García-Martínez, and M. Ladra, Universal enveloping crossed module of Leibniz crossed modules and representations, Journal of Physics: Conference Series 697 (2016), no. 1, 012007. [18] X. García-Martínez, A new characterisation of groups amongst monoids, Appl. Categ. Structures 25 (2017), no. 4, 659–661. [19] X. García-Martínez, E. Khmaladze, and M. Ladra, Non-abelian tensor product and homology of Lie superalgebras, J. Algebra 440 (2015), 464– 488. [20] X. García-Martínez and M. Ladra, Universal central extensions of Leibniz superalgebras over superdialgebras, Mediterr. J. Math. 14 (2017), no. 2, Art. 73, 15. [21] X. García-Martínez and M. Ladra, Universal central extensions of slpm, n, Aqof small rank over associative superalgebras, Turkish J. Math., 2017, doi:10.3906/mat-1604-3. [22] X. García-Martínez, R. Turdibaev, and T. Van der Linden, Do n-Lie algebras have universal enveloping algebras, J. Lie Theory 28 (2018), no. 1, 43–55. 257
[23] X. García-Martínez and T. Van der Linden, A characterisation of Lie algebras amongst alternating algebras, preprint arXiv:1701.05493, 2017. [24] X. García-Martínez and T. Van der Linden, A note on split extensions of bialgebras, preprint arXiv:1701.00665, 2017. [25] J. R. A. Gray, Algebraic exponentiation in general categories, Ph.D. thesis, University of Cape Town, 2010. [26] P. J. Higgins, Groups with multiple operators, Proc. London Math. Soc. (3) 6(1956), 366–416. [27] J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57–113. [28] S. A. Huq, Commutator, nilpotency, and solvability in categories, Quart. J. Math. Oxford Ser. (2) 19 (1968), 363–389. [29] G. Janelidze, L. Márki, and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002), no. 2-3, 367–386, Category theory 1999 (Coimbra). [30] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), no. 1, 139–158. [31] J.-L. Loday and T. Pirashvili, The tensor category of linear maps and Leibniz algebras, Georgian Math. J. 5(1998), no. 3, 263–276. [32] A. Montoli, D. Rodelo, and T. Van der Linden, Two characterisations of groups amongst monoids, J. Pure Appl. Algebra, 2016, doi:10.1016/j.jpaa.2017.05.005. [33] I. M. Musson, Lie superalgebras and enveloping algebras, Graduate Studies in Mathematics, vol. 131, American Mathematical Society, Providence, RI, 2012. [34] G. Orzech, Obstruction theory in algebraic categories. I, II, J. Pure Appl. Algebra 2(1972), 287–314; ibid. 2 (1972), 315–340. [35] A. R.-Grandjeán, Homología en categorías exactas, Ph.D. thesis, Universidade de Santiago de Compostela, 1970. 258
[36] G. S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195–222. [37] T. Van der Linden, Homology and homotopy in semi-abelian categories, Ph.D. thesis, Vrije Universiteit Brussel, 2006. 259