A new method for classifying complex filiform Lie algebras

A new method for classifying complex ®liform Lie algebras Luis Boza a , Eugenio M. Fedriani b , Juan N u~ nez b,* a Departamento de Matem atica Aplicada I, Escuela T ecnica Superior de Arquitectura, Universidad de Sevilla, Avda Reina Mercedes 2, 41012 Sevilla, Spain b Departamento de Geometr õa y Topolog õa, Universidad de Sevilla, Apartado 1160, 41080 Sevilla, Spain Abstract In this paper, we describe a new method to classify complex ®liform Lie algebras based on the concept of isomorphism between Lie algebras. This method, which has the advantage of being applied to any dimension, gives the families of algebras in each dimension in an explicit way. In order to apply, only the corresponding structure theorem of complex ®liform Lie algebras in each dimension is needed. As a consequence of our study, we also predict that the increase (in terms of quotiens) in the number of algebras families when passing from even dimension to odd dimension tends to 1 whereas it grows in a no ®nite way if passing from odd dimension to immediate even dimension. Ó2001 Elsevier Science Inc. All rights reserved. Keywords: Lie algebras; Filiform; Classi®cation; Isomorphism 1. Introduction Classifying Lie algebras is actually an open problem, which has been treated by several authors by using dierent methods. In fact, only the classi®cations of solvable, nilpotent and ®liform complex Lie algebras of dimension n, with n65, n67 and n612, respectively, are known. Applied Mathematics and Computation 121 (2001) 169±175 * Corresponding author. E-mail annresses: [email protected] (L. Blza), [email protected] (E.M. Felriani), [email protected] (l. Nu~nez). 0096-3003/01/$ - seel lnt matte PII: S 0 0 9 6 - 3 0 0 3 ( 9 9 ) 0 0 2 7 0 - 2 Dealing with the problem of classifying complex ®liform Lie algebras only, which is the aim of our study, the ®rst author who de®ned the ®liform Lie algebras was M. Vergne, in her thesis in 1966, which was later published in [10]. She ®rstly classi®ed these algebras in the cases of low dimensions and she also showed that they are a subset of the nilpotent Lie algebras (in fact, the most structured subset of nilpotent Lie algebras). In 1988, Goze and Ancochea [1], through the introduction of a new invariant which they call the characteristic sequence, obtained the classi®cation of these algebras in dimension 8. This classi®cation was later corrected by the same authors [2] and independently same time, by Seeley [9]. Later, Echarte and G omez [6] classi®ed using this method ®liform Lie algebras of dimension 9. Echarte and two of the authors of this paper classi®ed in 1996 these algebras in dimension 10 by the introduction of another invariant, which they call the pair i;h[4]. Recently, G omez et al. [7] by the method of the elementary changes of basis gave a correct classi®cation of complex ®liform Lie algebras of dimension n with n611. Finally, at the end of 1997, the last classi®cation of complex ®liform Lie algebras known until now appeared. The authors of this paper [5] classi®ed complex ®liform Lie algebras of dimension 12 by using this method which we explain in this paper, based on the concept of isomorphism between Lie algebras. However, this list, in which 496 families of these algebras appeared, had some errors which were checked by G omez et al. [7]. The authors thank them for their constructive suggestions in the application of this method. It is convenient to note that by classifying a set, we mean to ®nd a certain property of it which let us de®ne an equivalence relation among its elements. The equivalence relation which we will use to classify complex ®liform Lie algebras will be isomorphic. So, we will explicitly compute a representative of each class of complex ®liform Lie algebras in each dimension. However, this process requires hard and complicated calculations, which are impossible without the use of a computer. In our case, most of the calculations needed were made by using the Mathematica program, although another symbolic computation program could be used. 2. De®nitions and notations In this paper, all the Lie algebras which appear will be considered over the complex ®eld C. Let gCn;lbe a Lie algebra of dimension n, with lthe associated law. We consider the lower central series of gde®ned by C1gg;Ciglg;Ci1g. The Lie algebra gis ®liform if dimCCignifor 2 6i6n.Ifx2gwe denote by adxthe adjoint mapping associated to x(i.e., the map y7!lx;y). 170 L. Boza et al. / Appl. Math. Comput. 121 (2001) 169±175 Let gbe a ®liform Lie algebra of dimension n. Then there exists a basis Bfe1;...;engof gsuch that e12gnC2g, the matrix of ade1with respect to Bhas a Jordan block of order n1 and Cigis the vector space generated by fe2;...;eni1gwith 2 6i6n1. Such a basis is called an adapted basis. Sometimes, for the sake of simplicity, we will use x;yinstead of lx;yfor the Lie bracket in a Lie algebra. Let gand g0be two Lie algebras. A map U:g07!gis said to be a homomorphism between Lie algebras if Uis a linear application such that U:X;Y7!UX;UY 8X;Y2g0. In the case of being Ua bijection, it is called isomorphism. 3. A new method for classifying ®liform Lie algebras In order to classify complex ®liform Lie algebras of any dimension, we use the structure theorem of these algebras. So, let gbe a complex ®liform Lie algebra of dimension nwith an adapted basis fe1;...;eng. Then, e1;eheh1, 36h6n. A structure theorem for gconsists of giving explicitly the rest of the non-null brackets. These brackets are expressed as function of some structure constants aij. These constants, which de®ne the law of the algebra, are related themselves by polynomial relations coming from the Jacobi identities. These theorems are already known for dimension n,withn614 (see [3,4,8], for instance). To make this idea more understandable, we now present, for the sake of example, the structure theorem corresponding to the case of the dimension 8. If gC8;lis a ®liform Lie algebra,then there exists an adapted basis Bfe1;e2;...;e8gof gsuch that ·e1;eheh136h68; ·e4;e7a47e2; ·e5;e6a47e2; ·e4;e8a47e3a48e2; ·e5;e7a57e2; ·e5;e8a47e4a48 a57e3a58e2; ·e6;e7a57e3a67e2; ·e6;e8a47e5a48 2a57e4a58 a67e3a68e2; ·e7;e8a47e6a48 2a57e5a58 a67e4a68e3a78e2; where the rest of the brackets are null and the coecients a47;a48;...;a78 2C verify the following equation:a472a48 5a570. From now on,we say that a complex ®liform Lie algebra g,with basis fe1;e2;...;e8g,verifying the above condition,has the law la47;a48;a57;a58; a67;a68;a78,where the constants de®ne the law lof this algebra. L. Boza et al. / Appl. Math. Comput. 121 (2001) 169±175 171 As it can be seen in this example, the subindexes of these structure constants have been denoted according to the ®rst bracket in which they appear. In order to decide which is the ®rst bracket, we have set a certain order relation in the set of the subindexes pairs i;jwith i<j. This order is de®ned by i;jk;l()ij<kl_ijkl^i<k: In dimension n, the ®rst bracket we consider is e4;ena4ne2and we also use to order the constants in the law of the algebra. The main idea to classify these algebras is de®ning two Lie algebras and demanding them to be isomorphic. In this way, we get the necessary conditions for both to be isomorphic. In fact, sucient conditions are deduced when we classify each particular case, due to the isomorphisms between them. We obtain the classi®cation required from the dierent possibilities of some of the last conditions not to be veri®ed, that is, after grouping all the complex ®liform Lie algebras of each dimension in isomorphic classes. So, ®rst of all, we consider two bases over a same vector space verifying the thesis of the structure theorem of each dimension. Secondly, we set the existence of an isomorphism which maps the basis of the second algebra in the basis of the ®rst algebra. According to the de®nition of the isomorphism and to these theorems, we will get some restrictions, from which we can deduce the dierent families of algebras. In short, we give the following steps: Step 1. Let gand g0be two complex ®liform Lie algebras of dimension n, with bases fu1;...;ungand fv1;...;vng, respectively. From now on we denote the coecients de®ning their laws by aij and a0 ij. Let consider the isomorphism U:g0!g, this is, the bases change given by v1X n i1 siui;vnX n i1 riui and by the property of ®liformity of the algebra. Step 2. By straightforward computations in the previous step, we can deduce the following general expression: v2s1s1rnsnr1s1sna4;n1n4u2; where a4;n1is the coecient of un2in the bracket un1;un. In fact, it can be easily proved that a4;n10ifnis odd, whereas a4;n1is the ®rst structure constant with respect to the order previously de®ned, if nis even. Then, as v2belongs to a basis, necessarily v26 0 and thus, we have s16 0, s1rnsnr16 0 and s1sna4;n16 0: Step 3. Now, to make next the computations easier, by using again the property of ®liformity, we have 0 v3;vnr1s1rnsnr1s1sna4;n1n4 u2. Then, by taking into consideration the previous step, we have r10. So, 172 L. Boza et al. / Appl. Math. Comput. 121 (2001) 169±175 v1Pn i1siui;vnPn i2riui:We also deduce, in this step, the following conditions between the elements of the law of the algebra: s16 0;rn6 0 and s1sna4;n16 0: Step 4. As the bases change Uis an isomorphism between Lie algebras, then Uvk;Uvl  Uvk;vl  08k;l1;...;n. We obtain some vector equations by using these last expressions. Step 5. It is easy to prove in each dimension that we do not lose any piece of information although we operate with second components only. So, from now on, we will take into consideration these components only. Step 6. In this step, we already have new equations from the previous vector equations (which we denote cij 0) and the equations consisting on Pk0 (these last ones are the restrictions coming from the Jacobi identities in the corresponding structure theorem). So, we have a set of equations involving the coecients which appear in the laws of each algebra and the coecients riand sjfrom the isomorphisms. Now, we wish to solve the equation system constituted by the Pk0. It obliges us to distinguish several cases (for example, 16 of them are considered in dimension 12). By using the ®rst cij 0 we can easily prove that each case corresponds to families of algebras which are non-iso- morphic to the ones of other cases. Step 7. Finally, we wish that, in all of the cases of the previous step, the equations cij 0 give speci®c values (that is, constants) for the coecients of the law or of the isomorphisms or, in other case, relations between them. In this way, for instance, if we ®nd an expression like the following: a1s1a0 1, we would distinguish two non-isomorphic cases: a10a0 1and a16 06 a0 1(where a1and a0 1represent any coecient aij or a0 ij of the laws of g and g0, respectively). In this method, we use the following particular kind of sets to describe algebras having the same starting point in our method, in a non-redundant way and by using a unique family: Cmd dreiu:r>0;u20;2p m  and we denote Cm0Cm: Now, we will explain something more about these sets. The set Cmhas been already used in earlier classi®cations to solve some situations as the following. As we described in the general explanation, it is sometimes convenient to distinguish two cases depending on a20 or not. We now consider the second case. Then, let us suppose that we ®nd an expression like a2sm 1a0 20. Then, if we consider an isomorphism with s11=a0 21=m, any algebra of the family which we are considering is isomorphic to one with a21. So, from then on, we study the subfamily with a21. Then, we must suppose a21a0 2and we must also suppose that s1is a mth root of the unit. We can do it because this L. Boza et al. / Appl. Math. Comput. 121 (2001) 169±175 173 last condition will be veri®ed for any isomorphism between algebras belonging to that subfamily. If, when continuing the study of this same case, an equation of the kind a3s1a0 30 appears, we would have to distinguish two new cases: ·a30a0 3and s1is a mth root of the unit. ·a3s1a0 36 0. In this case, the sector Cmis sucient to represent in a unique way all possible values of a3of the algebras of that family, as a convenient value of s1relates both a3and a0 3by one isomorphism. So, from then on, we suppose a3a0 32Cmand s11. Something similar occurs with the set Cmd. There is no need for using these kinds of sets in dimensions less than 12, but it appears on some occasions for greater dimensions. For instance, let us suppose that we are using some expressions of the kind Aa0 4Ba4C0, where A;B;Care expressions depending on r12 (with r12 a square root of the unit), such that: if r12 1, it is deduced that a4a0 4and if r12 1, then a4a0 4d2C. That is, it occurs that either a4a0 4or d=2 is the middle point of a4and a0 4in the complex plane. Then, any possible value of a4in an algebra of the family which we are studying can be represented by one and only one element belonging to the set C2d=2. In this way, by applying our method we have obtained the classi®cation of complex ®liform Lie algebras of dimension nwith n612 and we have observed that our results coincide with the ones previously obtained by other authors for dimension 611, although they were reached by using dierent methods. So, we note that there is only a ®nite number of complex ®liform Lie algebras of dimension 6 pairwise non-isomorphic. In dimension 7 a no ®nite family of complex ®liform Lie algebras appears for the ®rst time. In dimension 8 some families have parameters not belonging to C, but either to C2or to another subset of Cde®ned by an easy algebraic restriction. In dimension 9 our method groups the algebras by using two two-parameter families and in the case of dimension 10, nine threeparameter families and the use of the set C3is needed. New diculties appear when applying our method to dimension 11. In this way, we ®nd four- and ®veparameter families of algebras. We also ®nd the appearance of some parameters belonging to C4and others appearing as solutions of third-grade equations. Finally, in dimension 12, the greatest dimension in which our method has been applied, we found the sets C5,C2dand C3d. Some equations with grades up to 6 containing parameters of exponent 4 were also used. With respect to the number of families obtained in each dimension when classifying complex ®liform Lie algebras, we have observed that this number (1 family of algebras in dimension 4, 2 in dimension 5, 5 in 6, 8 in 7, 22 in 8, 34 in 9, ...) allows us to predict that the increase (in terms of quotiens) in the number of algebras families when passing from even dimension to odd dimension (2=1;8=5;34=22;...is going to tend to 1 whereas it could grow in a no ®nite 174 L. Boza et al. / Appl. Math. Comput. 121 (2001) 169±175 way if passing from odd dimension to immediate even dimension (5=2; 22=8;104=34;.... This conjecture allows us to suppose that the classi®cation of complex ®liform Lie algebras of dimension 13 is not going to be very complicated with respect to that already obtained in dimension 12, whereas to get the classi®- cation in dimension 14 must result in more dicult and it must present quite a great complexity, not only in computations but in the number of families of algebras ®nally obtained. References [1] J.M. Ancochea, M. Goze, Classi®cati on des alg ebres de Lie ®liformes de dimension 8, Archiv. Math. 50 (1988) 511±525. [2] J.M. Ancochea, M. Goze, On the varietes of nilpotent Lie algebras of dimension 7 and 8, J. Pure Appl. Algebra 77 (1992) 431±440. [3] J.C. Benjumea, F.J. Castro, M.C. M arquez, J. N u~ nez, The equations of the sets of ®liform Lie algebras of dimension 13 and 14, Actas del 2°Encuentro de  Algebra Computacional y Aplicaciones (1996) 67±75. [4] L. Boza, F.J. Echarte, J. 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