Isomorphism and Isotopism Classes of Filiform Lie Algebras of Dimension up to Seven Over Finite Fields

arXiv:1510.07066v1 [math.RA] 23 Oct 2015 Isomorphism and isotopism classes of filiform Lie algebras of dimension up to seven O. J. Falc´on1, R. M. Falc´on2J. N´u˜nez3 1,3Faculty of Mathematics, University of Seville, Spain. 2School of Building Engineering, University of Seville, Spain. E-mail: [email protected], [email protected], [email protected] Abstract. Since the introduction of the concept of isotopism of algebras by Albert in 1942, a prolific literature on the subject has been developed for distinct types of algebras. Nevertheless, there barely exists any result on the problem of distributing Lie algebras into isotopism classes. The current paper is a first step to deal with such a problem. Specifically, we define a new series of isotopism invariants and we determine explicitly the distribution into isotopism classes of n-dimensional filiform Lie algebras, for n≤7. We also deal with the distribution of such algebras into isomorphism classes, for which we confirm some known results and we prove that there exist p+ 8 isomorphism classes of seven-dimensional filiform Lie algebras over the finite field Fpif p6= 2. Keywords: Filiform Lie algebra, isotopism, isomorphism. 2000 MSC: 17B60, 68W30. 1 Introduction. At present, the problem of the classification of Lie algebras in general is still unsolved. According to Levi’s theorem, there exist three types of Lie algebras: simple, solvable and semi-direct sum of semi-simple and solvable Lie algebras. Semi-simple Lie algebras include simple Lie algebras and solvable Lie algebras contain nilpotent Lie algebras. In turn, filiform Lie algebras constitute the most structured subset of nilpotent Lie algebras and have a large number of applications in Applied Mathematics, Engineering and Physics [12, 13]. They were introduced formally by Vergne [21, 22] in the late 1960s, although Umlauf had already used them as an example in his thesis [19]. The distribution into isomorphism classes of n-dimensional filiform Lie algebras over the complex field is known for n≤12 [6, 11], even if it is only known for nilpotent Lie algebras over the complex field of dimension n≤7 [4]. More recently, some authors have dealt with the classification of 1 n-dimensional nilpotent Lie algebras over finite fields Fp=Z/pZ. Specifically, Schneider [17] obtained the number of isomorphism classes over the finite field F2, for n≤9, and over F3and F5, for n≤7. The classification of six-dimensional nilpotent Lie algebras over a field of characteristic distinct of two was determined by de Graaf [14] and over any arbitrary field by Cical`o, de Graaf and Schneider [8]. With respect to the classification of filiform Lie algebras over Fp, Schneider obtained that Theorem 1.1 [17] There exist six six-dimensional filiform Lie algebras over F2and five over F3and F5;15 seven-dimensional filiform Lie algebras over F2,11 over F3and 13 over F5;47 eight-dimensional filiform Lie algebras over F2and 124 nine-dimensional filiform Lie algebras over F2. The main goal of this paper is to step forward in the distribution of filiform Lie algebras over any field, not only into isomorphism classes, which is the usual criterion, but also into isotopism classes, which is, at present, the first contribution on this subject. Isotopisms of filiform Lie algebras constitute a generalization of isomorphisms that can be used to gather together non-isomorphic algebras. The concept of isotopism of algebras was introduced by Albert [1] and developed later by Bruck [7], who introduced the concept of isotopically simple algebra as a simple algebra such that all their isotopic algebras are simple. He delved further into the study of isotopisms of division algebras and simple algebras. Isotopisms have been used since then to study and classify distinct algebraic structures like Jordan algebras [16], alternative algebras [5] or division algebras [18]. Nevertheless, there barely exists any result about isotopisms of Lie algebras, apart from the following two results of Albert and Bruck: Lemma 1.2 [1] A principal isotope gof a Lie algebra hwith respect to an isotopism (f, g, Id) is a Lie algebra if and only if the following two conditions are verified. i. [f(u), g(v)] = −[f(v), g(u)], for all u, v ∈g. ii. [f([f(u), g(v)]), g(w)] −[f([f(u), g(w)]), g(v)] −[f(u), g([f(v), g(w)])] = 0, for all u, v, w ∈g. 2 Theorem 1.3 [7] It is verified that: i. The Lie algebra of order n(n−1)/2, consisting of all skew-symmetric matrices, over any subfield of the field of all reals, under the multiplication A◦B=AB −BA, is isotopically simple. ii. The Lie algebra of order n(n−1), consisting of all skew-hermitian matrices in any field R(i)(where Ris a subfield of the reals and i2= 1), under the multiplication A◦B=AB −BA, is an isotopically simple algebra over R. More recently, Jim´enez-Gestal and P´erez-Iquierdo [15] have studied the relationship that exists between the isotopisms of a finite-dimensional real division algebra and the Lie algebra of its ternary derivations. On the other hand, Allison et al. [2, 3] have recently studied isotopes of a class of graded Lie algebras called Lie tori, but the notion of isotopism that they use is quite different from the conventional one. The structure of the paper is as follows: Section 2 is devoted to some preliminary concepts and results on Lie algebras that we use throughout the paper. We define a new series of isotopism invariants, which constitute the initial point of the distribution of six- and seven-dimensional filiform Lie algebras into isomorphism and isotopism classes that we determine, respectively, in Sections 3 and 4. In them, we also analyze those nonlinear equations that relate the structure constants of two isomorphic filiform Lie algebras of such dimensions with the entries of a matrix of an isomorphism between them. Such an analysis yields Theorem 1.1 and also the existence of p+ 8 isomorphism classes of seven-dimensional filiform Lie algebras, over Fp, for any prime p > 2. 2 Preliminaries. We start with some basic concepts and results on Lie algebras that we use throughout the paper. For a more comprehensive review on the subject, the reader can consult [20]. An n-dimensional Lie algebra gover a field Kis an n-dimensional vector space over Kendowed with a second inner law, named bracket product, that is bilinear and anti-commutative and that satisfies the Jacobi identity J(u, v, w) = [u, [v, w]] + [v, [w, u]] + [w, [u, v]] = 0,for all u, v, w ∈g.(1) 3 Given a basis {e1,...,en}of the Lie algebra g, the structure constants of g are the numbers ck ij ∈Ksuch that [ei, ej] = n X k=1 ck ijek,for 1 ≤i < j ≤n. (2) The centralizer of a subset hof gis the set Ceng(h) = {u∈g|[u, v] = 0,for all v∈h}. An ideal of gis any vector subspace h⊆gsuch that [h,g]⊆h. It is called abelian if [h,h] = 0. The center of gis the ideal Z(g) = Ceng(g). Given an m-dimensional ideal hof g, the quotient Lie algebra of gover his the quotient space g/h. It has structure of n−m dimensional Lie algebra with the product [u+h, v +h] = [u, v] + h,for all u, v ∈g.(3) Two Lie algebras gand g′are isotopic if there exist three regular linear transformations f, g and hfrom gto g′such that [f(u), g(v)] = h([u, v]),for all u, v ∈g.(4) It is denoted as g≃g′and it is said that g′is an isotope of g. The tuple (f, g, h) is called an isotopism of Lie algebras. If his the identity transformation Id, then the isotopism is said to be principal. If f=g=h, then it is an isomorphism of Lie algebras. It is denoted as g∼ =g′and the isomorphism (f, f, f) is denoted only as f. Given two isomorphic Lie algebras, gand g′, an isomorphism fbetween them and an ideal hof g, the vector subspace f(h) is an ideal of g′and the quotient Lie algebras g/hand g′/f(h) are also isomorphic. If h=Z(g), then f(h) = Z(g′). Hereafter, g(i)represents the quotient Lie algebra g(i−1)/Z(g(i−1)), for all natural i∈N, where g(0) =g. To be isotopic and isomorphic are equivalence relations on Lie algebras, which can then be partitioned into isotopism and isomorphism classes. Since isotopisms of Lie algebras constitute a generalization of isomorphism of such algebras, the problem of distributing Lie algebras into isotopism classes is reduced to that of gathering together isomorphism classes with isotopic equivalence class representatives. Even if there exist in the literature distinct isomorphism invariants that are traditionally used in the classification of Lie algebras into isomorphism classes, there does not exist any previous study about isotopism invariants. In the current paper, we define a sequence of such invariants, which also are, evidently, isomorphism invariants for Lie algebras. Specifically, given an n-dimensional Lie algebra gand a natural m≤n, we define dm(g) = max{dim Ceng(h): his an m-dimensional ideal of g}.(5) 4 The next result proves that the sequence d(g) = {d1(g),...,dn(g)}is an isotopism invariant. Lemma 2.1 Given two isotopic n-dimensional Lie algebras, gand g′, and a natural m≤n, it is dm(g) = dm(g′). Proof Let (f, g, h) be an isotopism between gand g′. Given m≤n, let hbe an m-dimensional ideal of gsuch that dm(g) = dim Ceng(h). Given x∈ Ceng(h), it is [f(x), g(h)] = h([x, h]) = h(0) = 0 and hence, f(Ceng(h)) ⊆ Ceng′(g(h)). Since fand gare regular transformations, it is dim g(h) = m and dm(g)≤dm(g′). The equality follows from a similar reasoning that makes use of the isotopism (f−1, g−1, h−1) between g′and g. The lower central series of a Lie algebra gis defined as the series of ideals C1(g) = g⊇ C2(g) = [g,g]⊇... ⊇ Ck(g) = [Ck−1(g),g]⊇... (6) If there exists a natural msuch that Cm(g)≡0, then gis called nilpotent. Nilpotency is preserved by isomorphisms of Lie algebras, because f(Ci(g)) = Ci(g′), for every isomorphism fbetween gand an isomorphic Lie algebra g′. The nil-index of gis the smallest natural psuch that Cp+1(g)≡0. Since Cp(g)⊆Z(g), the center of a non-trivial nilpotent Lie algebra is non-trivial. The type of gis defined as the sequence {dim g/C2(g),dim C2(g)/C3(g),...,dim Cp−1(g)/Cp(g)}.(7) According to Engel’s theorem, the Lie algebra gis isomorphic to a nilpotent Lie algebra of basis {e1,...,en}, such that ([e1, e2] = 0, [e1, ei] = ǫi−1ei−1,for all i∈ {3,...,n},(8) where ǫ2,...,ǫn−1∈ {0,1}. The Lie algebra gis said to be filiform if dim Ck(g) = n−k, for all k∈ {2,...,n}. Filiformity is also preserved by isomorphisms of Lie algebras. Further, the next two numbers were defined by Echarte et al. [9, 10] as isomorphism invariants for filiform Lie algebras. z1(g) = max{k∈N|Ceng(Cn−k+2(g)) ⊃ C2(g)}.(9) z2(g) = max{k∈N| Cn−k+1(g) is abelian}.(10) The only bidimensional filiform Lie algebra is the abelian. For n≥3, the type of an n-dimensional filiform Lie algebra gis {2,1,...,1}. Since the 5 center of gis one-dimensional, it is verified that d1(g) = nand dn(g) = 1. A basis {e1,...,en}of gis said to be compatible with respect to its lower central series if g2=he2,...,en−1i,g3=he2,...,en−2i,...,gn−1=he2i,gn= 0.(11) Hence, Z(g) = he2i(12) and [ei, ej]∈ he2,...,ei−1i,whenever 3 ≤i < j ≤n. (13) Since [e3, u]∈ he2iand J(e1, e3, u) = 0, for all u∈g, the fact of being e2∈Z(g) implies that [e3, u] = 0,for all u∈g2.(14) From (11), if n≥4, then [e4, en] = c2 4ne2+c3 4ne3. Thus, 0 = J(e1, e4, en) = [e1, c2 4ne2+c3 4ne3]−[e4, en−1] and hence, [e4, en−1]∈ he2i.(15) If the base field of an n-dimensional filiform Lie algebra ghas characteristic zero, then Vergne [21, 22] proved the existence of an adapted basis that facilitates the distribution of filiform Lie algebras into isomorphism classes. It can be considered, without loss of generality, to be compatible with respect to its lower central series, verifying Conditions (12-15), and can be chosen in such a way that the following conditions are also satisfied: ([e1, ei+1] = ei,for all i∈ {3,...,n−1}, [e3, en] = 0.(16) If the only brackets that are distinct of zero in an adapted basis are those of the form [e1, ei+1] = ei, then the corresponding filiform Lie algebra is called model. The n-dimensional model algebra is the only filiform Lie algebra having the isomorphism invariant z1equal to nand hence, it is not isomorphic to any other algebra of the same dimension. If the base field has characteristic distinct of zero, then it is not always possible to find an adapted basis of a filiform Lie algebra g. Due to it and similarly to the definition of a nilpotent basis proposed by Schneider in [17], we define a filiform basis of gas a compatible basis {e1,...,en}with respect to its lower central series such that, given i∈ {3,...,n}, it is [e1, ei] = ei−1 or [ei, en] = ei−1. The following results hold. 6 Lemma 2.2 Regardless of the base field, every finite-dimensional filiform Lie algebra has a filiform basis. Proof Let gbe a filiform Lie algebra over a field K. Since gis nilpotent, there exists a basis β={e1,...,en}satisfying Condition (8) and being compatible with respect to the lower central series of g. Let i∈ {3,...,n} be such that [e1, ei] = 0. From (11) and (13), there exists ai∈K\ {0}such that [aiei, en] = ei−1. We can suppose all these numbers aito be equal to 1. Otherwise, we replace each eiby aieiin the basis β. Proposition 2.3 The n-dimensional model algebra is not isotopic to any other filiform Lie algebra of the same dimension. Proof Given a non-model n-dimensional filiform Lie algebra gof filiform basis {e1,...,en}, there must exists a non-zero bracket [ei, ej], such that i6= 1 6=j. It implies that dn−1(g)< n −1. However, if his the ndimensional model algebra, then dm(h) = n−1, for all m∈ {2,...,n−1}. The result follows then from Lemma 2.1.  The only isomorphism class of the set of filiform Lie algebras of dimension n≤4 corresponds to the model algebra of such a dimension. It coincides, therefore, with the only isotopism class of such a set. For n= 5, there exist two isomorphism classes of filiform Lie algebras: the model algebra and that having an adapted basis satisfying the bracket [e4, e5] = e2. Both classes determine indeed distinct isotopism classes, because the isotopism invariant d4is equal to 4 for the model algebra, but it is equal to 2 for the algebras of the second class. For higher dimensions, the distribution into isomorphism and isotopism classes requires a more detailed study of the corresponding constant structures. We analyze them in the next sections, where we make use of the following lemma for the corresponding distribution into isomorphism classes. Lemma 2.4 Given a filiform Lie algebra gof filiform basis {e1,...,en}and an isomorphism fbetween gand a filiform Lie algebra g′, it is verified that f(ei)∈ Cn−i+1(g′), for all i∈ {2,...,n−1}. Proof The result follows straightforward from (11) and (13).  Given a filiform basis {e′ 1,...,e′ n}of g′, let F= (fij) be the regular square matrix that defines f, that is to say, such that f(ej) = Pn i=1 fije′ i, 7 for all natural j≤n. Since e2∈Z(g) and g2=he2,...,en−1i, the numbers f21 and f2ndo not have any influence on the isomorphism. We can suppose that f21 =f2n= 0 and hence, Lemma 2.4 yields that F=      f11 0 0 . . . 0f1n 0f22 f23 . . . f2(n−1) 0 f31 0f33 . . . f3(n−1) f3n . . . . . . . . ..... . . . . . f(n−1)1 0 0 . . . f(n−1)(n−1) f(n−1)n fn10 0 . . . 0fnn       (17) 3 Classification of six-dimensional filiform Lie algebras. The distribution into isomorphism classes of six-dimensional filiform Lie algebras is known over any field [8, 14, 17]. Nevertheless, it is interesting to deal again with such a distribution, independently of the known results, in order to take advantage of such a small dimension to expose in detail the algebraic methodology related to the new isotopism invariants. Since the reasoning is analogous for higher dimensions, we avoid in the following section to show explicitly all the steps, because of their extensive and tedious computation. A case study of possible six-dimensional filiform bases allows us to assure the next result. Lemma 3.1 Given a six-dimensional filiform Lie algebra gover a field K, there exist three numbers a, b, c ∈Kand an adapted basis of gsuch that g∼ =g6 abc ≡           [e1, ei+1] = ei,for all i > 1, [e4, e5] = ae2, [e4, e6] = be2+ae3, [e5, e6] = ce2+be3+ae4. (18)  The isotopism invariants of the previous algebras determine a preliminary classification for their distribution into isotopism and isomorphism classes. d(g6 abc) =            {6,5,5,5,5,1},if a=b=c= 0, {6,5,3,2,2,1},if a=b= 0 6=c, {6,5,4,4,2,1},if a= 0 6=b, {6,5,3,2,2,1},if a6= 0. (19) 8 Lemma 2.1 yields then the existence of at least four isotopism (isomorphism) classes of six-dimensional filiform Lie algebras over any field. Since the model algebra g6 000 constitutes an isotopism (isomorphism) class by itself, we focus on the rest of cases. We start with the distribution into isomorphism classes. Proposition 3.2 Given c, C ∈K\{0}, it is g6 00c∼ =g6 00C. Proof It is enough to consider the isomorphism fbetween them, defined as K-linear extension of f(ei) = (e1,if i= 1, c Cei,otherwise.  The remaining cases require an analysis of the entries of the matrix F= (fij) of an isomorphism between two six-dimensional isomorphic filiform Lie algebras over a field K,g6 abc and g6 ABC, of respective adapted bases {e1,...,e6}and {e′ 1,...,e′ 6}. Its form has been exposed in (17). Further, since −f33f16e′ 2= [f(e3), f(e6)] = f([e3, e6]) = f(0) = 0 and Fis regular, it must be f16 = 0. The central columns of Fcan be determined by using the entries of the first and last columns. Lemma 3.3 It is verified that a) f55 =f11f66. b) f44 = (f11 −Af61)f55. c) f33 = (f11 −Af61)f44. d) f22 =f11f33. e) f45 = (f11 −Af61)f56 +Af51f66. f) f34 = (f11 −Af61)f45 −Bf55f61. g) f23 =f11f34 −(Af51 +Bf61)f44. h) f35 = (f11 −Af61)f46 +Af41f66 +B(f51f66 −f56f61). i) f24 =f11f35 −(Af51 +Bf61)f45 + (Af41 −Cf61)f55. j) f25 =f11f36 +A(f41f56 −f46f51)+B(f41f66 −f46f61)+C(f51f66 −f56f61). 9 Hence, the algebras g7 a0cd and g7 A0CD are isomorphic if and only if one of the following conditions is satisfied i. 4ad = 5c2and 4AD = 5C2. ii. 4ad 6= 5c2, 4AD 6= 5C2and (4AD −5C2)/(4ad −5c2) is a perfect square in K. Assertions (b) and (c) follow then immediately from (i) and (ii). Finally, let pbe a prime distinct of two and let qbe a non-perfect square in K=Fp. Every perfect square rin Fpis uniquely related to a non-perfect square s in Fpsuch that r=q/s. Such a relation is 1-1, because Fpis a finite field that contains exactly (p−1)/2 perfect squares distinct of zero. Assertion (d) follows then from (ii).  Proposition 4.5 Let g7 abcd and g7 ABCD be two non-model algebras such that a, b, A and Bare all of them distinct of zero. They are isomorphic if and only aB =Ab and one of the following assertion is verified 1. The characteristic of the base field is distinct of two and a+b6= 0. 2. If the characteristic of the base field is two or a+b= 0, then it must be c= 0 = Cor c6= 0 6=C. Proof If the characteristic of the base field is distinct of two, then we can isolate f57 from Equation (37). Otherwise, we isolate f67. Further, since A6= 0, we have from (34) that f77 =af2 11/A =bf2 11/B and hence, aB =Ab. Equation (36) yields then that aAcf2 11 −a2Cf11 = 2(a+b)2A2f71. If the characteristic of the base field is distinct of two and a+b6= 0, then a possible isomorphism can be determined once f71 is isolated from the previous equation. Otherwise, it is Acf11 =aC and hence, there exists an isomorphism between g7 abcd and g7 A(Ab/a)CD only if c=C= 0 or c6= 0 6=C. In the last case, it is enough to consider f11 =aC/Ac. The previous results establish the distribution into isomorphism classes of seven-dimensional filiform Lie algebras. 16 Theorem 4.6 It is verified that a) There exist 15 isomorphism classes of seven-dimensional filiform Lie algebras over a field of characteristic two: g7 0000,g7 0001,g7 0010,g7 0011,g7 0100,g7 0110,g7 1000,g7 1010, g7 1011,g7 1100,g7 1110,g7 0,g7 1,h7 0and h7 1. b) Any seven-dimensional filiform Lie algebra over an algebraically closed field of characteristic distinct of two is isomorphic to one of the following isomorphism classes: g7 0000,g7 0001,g7 0010,g7 0011,g7 0100,g7 1001,g7 1b00,and g7 1(−1)10, where b∈K. c) Given a prime p6= 2, there exist p+ 8 isomorphism classes of sevendimensional filiform Lie algebras over the finite field Fp: g7 0000,g7 0001,g7 0010,g7 0011,g7 0100,g7 1001,g7 100q,g7 1b00 and g7 1(−1)10, where b∈Kand qis a non-perfect square of Fp. Finally, the distribution of seven-dimensional filiform Lie algebras into isotopism classes is determined in the next result. Proposition 4.7 It is verified that a) There exist 10 isotopism classes of seven-dimensional filiform Lie algebras over a field of characteristic two: g7 0000,g7 0001,g7 0010,g7 0100,g7 1000,g7 1100,g7 1110,g7 0,g7 1and h7 0. b) There exist eight isotopism classes of seven-dimensional filiform Lie algebras over an algebraically closed field of characteristic distinct of two and also over the finite field Fp, where pis a prime distinct of two: g7 0000,g7 0001,g7 0010,g7 0100,g7 1000,g7 1100,g7 1(−1)00 and g7 1(−1)10. Proof We have already seen that h7 0≃h7 1and that the algebras g7 0and g7 1constitute two distinct isotopism classes by themselves. We determine also the following isotopisms, which are defined as linear extensions from the exposed non-trivial changes of the corresponding bases: 17 a) g7 0011 ≃g7 0010, by the isotopism (f, f, h), where f(e4) = e4−e3and h(e3) = e3−e2. b) g7 0110 ≃g7 0100, by the isotopism (f, f, h), where f(e1) = e1+e7, f(e7) = e6+e7, h(e4) = e4−e3, h(e5) = e5−e4and h(e6) = e6+e5−e4. c) g7 10cd ≃g7 1000, for all c, d ∈K, by the isotopism (f, f, h), where f(e5) = e5+ (c2−d)e3, h(e3) = e3−ce2, h(e4) = e4−ce3+ (c2−d)e2and h(e5) = e5−ce4. d) g7 1b00 ≃g7 1100, for all b∈ {2,...,p−2}, by the isotopism (f, f, h), where f(e3) = f44e3, f(e4) = f44e4, f(e5) = f55e5, f(e6) = f66e6, h(e2) = f44e2, h(e3) = f44e3, h(e4) = f55e4, h(e5) = f66e5, with f44 =4 (b+ 1)2f66, f55 =2 b+ 1f66, f66 =2b b+ 1 A simple analysis on possible isotopisms among the Lie algebras g7 1100, g7 1(−1)00 and g7 1(−1)10 indicates that they determine distinct isotopism classes. The result follows then from Lemma 2.1 and the invariants (27)-(29).  5 Final remarks and further work. In the current paper we have dealt with the distribution of Lie algebras into isotopism classes. We have defined a series of isotopism invariants of Lie algebras, which we have used to determine the isotopism and isomorphism classes of filiform Lie algebras of dimension n≤7. We have generalized the study of Schneider [17] since we have obtained that there exist p+ 8 isomorphism classes of seven-dimensional filiform Lie algebras over the finite field Fp, for any prime p6= 2. Higher dimensions can be similarly analyzed, although with more extensive case studies. On the other hand, it remains open the problem of distributing into isotopism classes other families of Lie algebras, distinct of the filiform case. 18 References [1] A. A. Albert, Non-Associative Algebras: I. Fundamental Concepts and Isotopy, Ann. of Math., Second Series 43:4 (1942), 685–707. [2] B. Allison, S. Berman, J. 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