Description of some families of filiform Lie algebras

Houston Journal of Mathematics c °2008 University of Houston Volume 34, No. 1, 2008 DESCRIPTION OF SOME FAMILIES OF FILIFORM LIE ALGEBRAS F. J. ECHARTE, J. N´ U˜ NEZ, AND F. RAM´ IREZ Communicated by the editors Abstract. In this paper we describe some families of filiform Lie algebras by giving a method which allows to obtain them in any arbitrary dimension n starting from the triple (p, q, m), where m=nand pand qare, respectively, invariants z1and z2of those algebras. After obtaining the general law of complex filiform Lie algebras corresponding to triples (p, q, m), some concrete examples of this method are shown. 1. Introduction Sometimes, people who work on Lie Theory need dispose of lots of examples of Lie algebras of high dimensions to check if some property verified by Lie algebras of low dimensions is also satisfied in them. These examples are also needed, for instance, to verify if a previous conjecture, which has been observed in Lie algebras of low dimensions, continues being true when the dimension increases. However, at present, it is difficult to have those examples of Lie algebras of dimension, say, non-excessively low. Indeed, if we do not take into consideration simple and semisimple Lie algebras, whose classification was obtained by Killing, Cartan and others, in the last decade of the last next century, only the classifications of solvable, nilpotent and filiform complex Lie algebras of dimension n, with n≤5, n≤7 and n≤12,respectively, are actually known (see [1], [10] and [4], for instance). Moreover, this problem worsens if we note that not only the classification problem of Lie algebras is still unsolved, but, in fact, it seems to be accepted by experts (like Shalev and Zelmanov [13]) that one will not totally classify Lie 2000 Mathematics Subject Classification. 17B30. Key words and phrases. Lie algebra, filiform, invariant. 19 20 ECHARTE, N´ U˜ NEZ, AND RAM´ IREZ algebras of finite dimension (see also Goze-Khakimdjanov [11]). Indeed, it is reasonable for research workers in this theory to impose further conditions to these algebras to obtain, at least, partial results. In this sense, the main goal of this paper is to completely describe a family of complex filiform Lie algebras in any dimension ndefined by some invariants (introduced by us in [7]) computed on centralizer of some characteristic ideal of its descending central sequence. Now, we will briefly explain why we deal with complex filiform Lie algebras. These algebras were introduced by M. Vergne in the late 60’s of the past century [14]. However, before that, Blackburn studied the analogous class of finite pgroups and used the term maximal class to call them, which is also now used for Lie algebras [2]. In fact, both terms filiform and maximal class are synonymous. Vergne showed that, within the variety of nilpotent Lie multiplications on a fixed vector space, non-filiforms can be relegated to small-dimensional components; thus, from some intuitive point of view, it is possible to consider that quite a lot nilpotent Lie algebras are filiform, in spite of this last subset not being dense in the space of nilpotent Lie algebras. Apart from that, complex filiform Lie algebras are the most structured subset of nilpotent Lie algebras, with respect to an adapted basis (see next section). In this sense, we can study and classify them easier than the set of nilpotent Lie algebras. So, in earlier papers, some of us have deeply studied these algebras and obtained quite a lot results about them. Indeed, we already got the classification of those having dimensions 10,11 and 12 (see [3], [10] and [4]), for instance). Moreover, we think that this study can be also considered a little step forward in the problem of the classification of these algebras, although it is not the principal objective of this work. 2. Definitions and notations A nilpotent Lie algebra gis said to be filiform, if it is verified that: dim g2=n−2; . . . dim gk=n−k;. . . dimgn= 0, where dim g=n, and gk= [g,gk−1], 2 ≤k≤n. A basis {e1, . . . , en}of gis called an adapted basis if: [e1, e2] = 0; [e1, eh] = eh−1(h= 3, . . . , n),[e3, en] = 0. Note that the definition of filiformity assures that every filiform Lie algebra has an adapted basis. SOME FAMILIES OF FILIFORM LIE ALGEBRAS 21 It is easy to deduce that, with respect to this basis, it is verified: g2≡ {e2, . . . , en−1},g3≡ {e2, . . . , en−2}, . . . , gn−1≡ {e2},gn≡ {0}. A filiform Lie algebra gis said to be model, if the only non-null brackets between the elements of an adapted basis are: [e1, eh] = eh−1(h= 3, . . . , n).When the dimension is given, this model is unique up an isomorphism and every filiform Lie algebra is a deformation of the model (see Goze-Khakimdjanov [11]). We note by Cghthe centralizer of a subalgebra hin g. In [6] (although by using a different notation for denoting it, which was later improved), two of the authors of this paper defined the integer: z1=z1(g) = max{k∈N| Cg(gn−k+2)⊃g2}. Note that this definition means that the ideal gn−z1+2 is the greatest ideal whose centralizer contains g2, that is, the ideal whose centralizer is the ideal g, generated by {e2, . . . , en−1, en}. Note as well that, according to this last definition, z1(g) is an invariant of filiform Lie algebras. In terms of adapted basis, it is deduced in [6] that: z1=min{k∈N− {1} | [ek, en]6= 0}. However, note that z1(g) does not always exist. In this case, it is also easy to see that gis a model algebra. It implies that the algebra with basis {e2, . . . , en} is commutative. It allows us to give a new definition of model algebra, which is also independent of any adapted basis: we say that a filiform Lie algebra g is a model algebra if Cg(gn−2) is commutative. Both definitions are equivalent, since gn−2is the ideal with basis {e2, e3}whose centralizer is the ideal with basis {e2, . . . , en}. In [7] (a different notation for denoting it was also used) the integer z2(g) was introduced by us as follows: z2=z2(g) = max{k∈N|gn−k+1 is commutative} Note that this definition means that the ideal gn−z2+1 ≡ {e2, . . . , ez2}is the greatest commutative subalgebra in the nilpotency sequence. In that paper we also proved the three following asserts: a) that z2(g) is an invariant of complex filiform Lie algebras. b): that there exists at least some bracket [ek, ek+1]6= 0, for some k < n, in every non-model complex filiform Lie algebra of dimension nand c): that z2(g) = min {k∈N|[ek, ek+1]6= 0}. 22 ECHARTE, N´ U˜ NEZ, AND RAM´ IREZ Similarly as the case of z1(g), if the set of this definition is empty, then gis a model algebra. In the other case, the smallest value of z2(g) is 4, because of being [e1, e2]=[e2, e3] = [e3, e4] = 0 in any adapted basis. With respect to both invariants, the following two results are verified [9]: Lemma 1. [ez1+k−1, ez2+1] = α1ek+1 +α2ek+. . . +αk−1e3+αke2,with 1≤k≤z2−z1+ 1. Moreover, αp6= 0 for some psuch that 1≤p≤z2−z1+ 1. Lemma 2. [ez1, ez2+k] = β1ek+1 +β2ek+. . . +βk−1e3+βke2, with 1≤ k≤n−z2(where β1=α1from Lemma 1). Moreover, βq6= 0 for some qsuch that 1≤q≤n−z2. Finally, it is also proved in [7] the following relation among invariants z1,z2 and nof complex filiform Lie algebras: 4≤z1≤z2< n ≤2z2−2 which will be used later. From now on, we will suppose that all the Lie algebras appearing in this paper are complex filiform and that all bases are adapted. We denote by J(a, b, c) = 0 Jacobi identity associated with vectors a,band c. 3. Filiform Lie algebras verifying z2(g) = n−2 In this section we study n-dimensional filiform Lie algebras satisfying z2(g) = n−2.Note that in these algebras, n≥6, since z2≥4 is always verified. Due to the filiformity and to definitions of invariants z1and z2, the only nonnull brackets are [e1, eh] = eh−1, for all 3 ≤h≤n, and [eh, ek], with z1≤h < k≤n−1 (obviously, [ek, eh] = −[eh, ek], if h < k). Remember that, according Lemma 1, we have the following expressions, where α1could be equal to 0. [ez1, en−1] = α1e2 [ez1+1, en−1] = α1e3+α2e2 . . .. . . [en−3, en−1] = α1en−z1−1+α2en−z1−2+. . . +αn−z1−3e3+αn−z1−2e2 [en−2, en−1] = α1en−z1+α2en−z1−1+. . . +αn−z1−2e3+αn−z1−1e2 in which at least, some αp(with 1 ≤p≤n−z1−1) is necessarily non-null (see Lemma 1). Then, by taking into consideration Lemma 2, if we denote β2=γ2, we have [ez1, en−1] = α1e2and [ez1, en] = α1e3+γ2e2and thus, at least one of both, α1 or γ2is non-null. SOME FAMILIES OF FILIFORM LIE ALGEBRAS 23 Now, from Jacobi identity J(e1, ez1+1, en) = 0,we deduce: [ez1+1, en] = 2 α1e4+ (α2+γ2)e3+γ3e2 In the same way, from Jacobi identities J(e1, ez1+2, en) = 0 and J(e1, ez1+3, en)=0,we have, respectively: [ez1+2, en] = 3 α1e5+ (2 α2+γ2)e4+ (α3+γ3)e3+γ4e2,and [ez1+3, en] = 4 α1e6+ (3 α2+γ2)e5+ (2 α3+γ3)e4+ (α4+γ4)e3+γ5e2. So, by proceeding in the same way, we can deduce (when the first subindex is z1+ (n−z1+ 3) ≡n−3) that [en−3, en]=(n−z1−2) α1en−z1+ ((n−z1−3) α2+γ2)en−z1−1+ ((n−z1−4) α3+γ3)en−z1−2+. . . + (3 αn−z1−4+γn−z1−4)e5+ (2 αn−z1−3+γn−z1−3)e4+ (αn−z1−2+γn−z1−2)e3+γn−z1−1e2 By replacing now these two results previously obtained in Jacobi identity J(e1, en−2, en)=0,we have that [e1,[en−2, en] = (n−z1−1) α1en−z1+ ((n− z1−2) α2+γ2)en−z1−1+ ((n−z1−3) α3+γ3)en−z1−2+. . . + (3 αn−z1−3+ γn−z1−3)e4+ (2 αn−z1−2+γn−z1−2)e3+ (αn−z1−1+γn−z1−1)e2and thus: [en−2, en] = (n−z1−1) α1en−z1+1 + ((n−z1−2) α2+γ2)en−z1+ ((n−z1−3) α3+γ3)en−z1−1+. . . + (3 αn−z1−3+γn−z1−3)e5+ (2 αn−z1−2+γn−z1−2)e4+ (αn−z1−1+γn−z1−1)e3+γn−z1e2. Finally, by using Jacobi identity J(e1, en−1, en) = 0 and by proceeding in the same way we deduce that: [en−1, en] = (n−z1−1) α1en−z1+2 + ((n−z1−2) α2+γ2)en−z1+1+ ((n−z1−3) α3+γ3)en−z1+. . . + (3 αn−z1−3+γn−z1−3)e6+ (2 αn−z1−2+γn−z1−2)e5+ (αn−z1−1+γn−z1−1)e4+γn−z1e3+γn−z1+1 e2. Therefore, the following result has been proved: 24 ECHARTE, N´ U˜ NEZ, AND RAM´ IREZ Theorem 3.1. (Main Theorem) The laws of filiform Lie algebras verifying z2(g) = n−2, with respect to an adapted basis {e1, . . . , en}, are the following: [e1, eh] = eh−1(3 ≤h≤n) [ez1, en−1] = α1e2 [ez1, en] = α1e3+γ2e2(with either α16= 0 or γ26= 0) [ez1+1, en−1] = α1e3+α2e2 [ez1+1, en] = 2 α1e4+ (α2+γ2)e3+γ3e2 [ez1+2, en−1] = α1e4+α2e3+α3e2 [ez1+2, en] = 3 α1e5+ (2 α2+γ2)e4+ (α3+γ3)e3+γ4e2 [ez1+3, en−1] = α1e5+α2e4+α3e3+α4e2 [ez1+3, en] = 4 α1e6+ (3 α2+γ2)e5+ (2 α3+γ3)e4+ (α4+γ4)e3 +γ5e2 [ez1+4, en−1] = α1e6+α2e5+α3e4+α4e3+α5e2 [ez1+4, en] = 5 α1e7+ (4 α2+γ2)e6+ (3 α3+γ3)e5+ (2 α4+γ4)e4+ (α5+γ5)e3+γ6e2 . . .. . . [en−3, en] = (n−z1−2) α1en−z1+ ((n−z1−3) α2+γ2)en−z1−1+ ((n−z1−4) α3+γ3)en−z1−2+. . . + (3 αn−z1−4+γn−z1−4)e5+ (2 αn−z1−3+γn−z1−3)e4+ (αn−z1−2+γn−z1−2)e3γn−z1−1e2 [en−2, en−1] = α1en−z1+α2en−z1−1+α3en−z1−2+. . . +αn−z1−3e4+ αn−z1−2e3+αn−z1−1e2(with αp6= 0, for at least some p, with 1≤p≤n−z1−1) [en−2, en] = (n−z1−1) α1en−z1+1 + ((n−z1−2) α2+γ2)en−z1+. . . + (4 αn−z1−4+γn−z1−4)e6+ (3 αn−z1−3+γn−z1−3)e5+ (2 αn−z1−2+γn−z1−2)e4+ (αn−z1−1+γn−z1−1)e3+γn−z1e2 [en−1, en] = (n−z1−1) α1en−z1+2 + ((n−z1−2) α2+γ2)en−z1+1+ ((n−z1−3) α3+γ3)en−z1 +. . . + (4 αn−z1−4+γn−z1−4)e7+ (3 αn−z1−3+γn−z1−3)e6+ (2 αn−z1−2+γn−z1−2)e5+ (αn−z1−1+γn−z1−1)e4+αn−z1e3+γn−z1+1 e2 where the coefficients αpand γq, with 1≤p≤n−z1and 2≤q≤n−z1+ 1 are abided by the restrictions derived from Jacobi identities J(eh, ek, el) = 0, with z1≤h < k < l ≤n−2.¤ So, as either α16= 0 or γ26= 0,we now distinguish two cases in the study of these algebras: SOME FAMILIES OF FILIFORM LIE ALGEBRAS 25 •The case α1= 0. In this case, γ26= 0.Moreover, a suitable adapted base change involves γ2= 1. Besides, as the bracket [en−2, en−1]6= 0,we deduce that there exists at least a coefficient αpdifferent from 0, with 2 ≤p≤n−z1−1. Example 1. To illustrate this case, let us suppose that we would like to find an example of a filiform Lie algebra corresponding to the triple (4,5,7). To do this, by replacing z1,z2and kwith 4,5and 7, respectively, in Theorem 3.1, we obtain the following expressions: [e1, eh] = eh−1(3 ≤h≤7) [e4, e6] = α1e2 [e4, e7] = α1e3+γ2e2 [e5, e6] = α1e3+α2e2 [e5, e7] = 2 α1e4+ (α2+γ2)e3+γ3e2 [e6, e7] = α1e4+α2e3+α3e2 [e6, e7] = 3 α1e5+ (2 α2+γ2)e4+ (α3+γ3)e3+γ4e2. So, as γ26= 0 in this case, we can suppose α1= 0 and γ2= 1 (remember that a suitable adapted base change involves γ2= 1 and that the subject of possible isomorphisms between Lie algebras which are obtained is not being dealt in this paper). After, we can take, for instance, the following values for the rest of coefficients: γ3=−α2= 1 and γ4= 0. Then, in this way, we would obtain the following filiform Lie algebra (with respect to the adapted basis {e1, . . . , e7}): [e1, eh] = eh−1(3 ≤h≤7) [e4, e7] = e2, [e5, e6] = −e2,[e5, e7] = e2,[e6, e7] = e3, So, it can be checked that every filiform Lie algebra satisfying n= 7, z1= 4, z2= 5 is isomorphic to the one named µ4 7by Ancochea and Goze’s classification of nilpotent Lie algebras of dimension 7(see [1]). However, since this classification there exist many other true classifications. For example, the reader can consult the one by Goze and Remm, which is available en the web site [12]. •The case α16= 0. In this case, it is always possible to suppose α1= 1.It can be obtained by using a suitable adapted basis change. Example 2. To show an example of this case, let us suppose that we would like to find an example of a filiform Lie algebra corresponding to the triple (6,7,9). By proceeding in the same way as in previous example, we have the following 26 ECHARTE, N´ U˜ NEZ, AND RAM´ IREZ expressions: [e1, eh] = eh−1(3 ≤h≤9) [e6, e8] = α1e2 [e6, e9] = α1e3+γ2e2 [e7, e8] = α1e3+α2e2 [e7, e9] = 2 α1e4+ (α2+γ2)e3+γ3e2 [e8, e9] = 3 α1e5+ (2 α2+γ2)e4+ (α3+γ3)e3+γ4e2. So, as it is possible to get α1= 1 in this case, we can suppose γ2= 1, for instance. Next, by taking α2=λ(undetermined) and γ3=γ4=α3= 0,we obtain the following filiform Lie algebra, with respect to the adapted basis {e1, . . . , e9}: [e1, eh] = eh−1(3 ≤h≤9) [e6, e8] = e2, [e6, e9] = e3+e2,[e7, e8] = e3+λ e2, [e7, e9] = 2 e4+ (1 + λ)e3,[e8, e9] = 2 e5+ (1 + λ)e4 As it can be also checked, this family of algebras is isomorphic with the one named µ16,λ 9by Echarte and Gomez’s classification of filiform Lie algebras of dimension 9(see [5]). Note that this procedure can be implemented in any symbolic computation package, which allows to obtain concrete examples of filiform Lie algebras in any dimension in an easy way. However, note also that a filiform Lie algebra can be obtained by two cases, as we can observe in the following table (see the triple (5, 6, 8)): triple bracket case triple bracket case (4,4,6) [e4, e5] = e22 (5,6,8) [e5, e7] = 0 1 (4,5,7) [e4, e6] = 0 1 (5,6,8) [e5, e7] = e22 (5,5,7) [e5, e6] = e22 (6,6,8) [e6, e7] = e22 (4,6,8) [e4, e7] = 0 1 Now, we are going to study some particular cases of these algebras corresponding to the triple (z1, n −2, n).From this study, we will obtain some consequences which will allow us to establish some conjectures. These cases are the following: SOME FAMILIES OF FILIFORM LIE ALGEBRAS 27 3.1. The case z1=n−2.In this case, there is an unique coefficient α1, which is α16= 0 (in fact, we can get α1= 1, as we said before). Then, these algebras are defined, with respect to an adapted basis {e1, . . . , en}, by brackets: [e1, eh] = eh−1(3 ≤h≤n) [en−2, en−1] = e2 [en−2, en] = e3+γ2e2 [en−1, en] = e4+γ2e3+γ3e2 It is easy to check that each addend of Jacobi identity J(en−2, en−1, en) = 0 is null. Besides, it can be also observed that the adapted base change: e0 k=ek(1 ≤k≤3) ; e0 h=eh+γ3 2eh−2(4 ≤h≤n) involves γ0 3= 0.These results allow to set the following: Proposition 1. With respect to an adapted basis {e1, . . . , en}, filiform Lie algebras belonging to the triple (n−2, n −2, n)are defined by: [e1, eh] = eh−1(3 ≤h≤n) [en−2, en−1] = e2 [en−2, en] = e3+γ2e2 [en−1, en] = e4+γ2e3 with γ2∈C.¤ Note that in the particular cases n= 6,7,8, this result had been already obtained in [8]. Moreover, if we use a suitable basis change in these cases, it can be observed that if nis 7, that is, odd, then γ2= 0, whereas if nis even (6 or 8), then that coefficient can be or not be equal to zero. We conjecture that it also occurs for any narbitrary. Now, we are going to consider a new case of these algebras: 3.2. The case z1=n−3.The filiform Lie algebras corresponding to this case are defined, with respect to an adapted basis {e1, . . . , en}, by the brackets: [e1, eh] = eh−1(3 ≤h≤n) [en−3, en−1] = α1e2 [en−3, en] = α1e3+γ2e2(α16= 0 or γ26= 0) [en−2, en−1] = α1e3+α2e2(α16= 0 or α26= 0) [en−2, en] = 2 α1e4+ (α2+γ2)e3+γ3e2 [en−1, en] = 2 α1e5+ (α2+γ2)e4+γ3e3+γ4e2 where coefficients are abided to restrictions resulting from Jacobi identity J(en−2, en−1, en)=0.