Filiform Lie algebras with low derived length
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Filiform Lie algebras with low derived length F.J. Castro-Jiménez, M. Ceballos, J. Núñez-Valdés 1 Abstract We construct, for any n≥5 , a family of complex liform Lie algebras with derived length at most 3 and dimension n . We also give examples of n -dimensional liform Lie algebras with derived length greater than 3 . Keywords: Filiform Lie algebra, derived length, Lie algebra invariants. 2010 Mathematics Subject Classication: 17B30, 1708, 17B05, 68W30. 1 Introduction The derived length, also known as solvability index, of nilpotent groups or nilpotent Lie algebras has been studied by a number of authors. In the case of nite p groups this study was initiated by Burnside [11, 12] and then continued by several authors (e.g. Philip Hall [22], Magnus [27], Itô [24], Hall-Higman [23], Blackburn [4], Mann [28] among many others). In the case of a nilpotent Lie algebra, the study of its derived length has been treated in several papers, e.g. by Dixmier [15], Patterson [31, 32], Bokut [5]. In [6] the authors show that there are liform Lie algebras of arbitrary derived length. They describe, for any k≥2 , a liform Lie algebra of derived length k and dimension n for each n satisfying 2k≤n+ 1 <2k+1 . These algebras were studied by Benoist in [3]. In this paper we construct, for any n≥5 , a family of complex liform Lie algebras with derived length at most 3 and dimension n , see Theorem 3. To this end we improve the result [14, Prop. 2] and describe the law of any liform Lie algebra of dimension n≥5 with respect to a suitable adapted basis by using two numerical invariants associated with the algebra, see Theorem 2. This result is related to the description of the ane variety of n dimensional liform Lie algebras given in [29, Sec. 4], where the variety of Lie algebras of maximal class is described. See also [34] where the related notion of narrow Lie algebras is treated. Theorem 2 generalizes a result of Bratzlavsky [7] valid for metabelian Lie algebras. We also construct a family of complex liform Lie algebras of dimension 15 and derived length 4 generalising the one given by [8, Ex. 3.2]. Let us remark that there is no complex liform Lie algebra of dimension less than or equal to 14 and derived length 4 (see [8, Prop. 3.7]). If the derived length of a solvable Lie algebra is k one also says that the algebra is k -step solvable, see Denition 2. Solvable Lie algebras with low derived length have been previously 1 FJCJ: Departamento de Álgebra e IMUS. Facultad de Matemáticas, Universidad de Sevilla. C/ Tara s/n, 41012 Seville (Spain). [email protected] MC: Dpto. de Ingeniería. Universidad Loyola Andalucía, Campus Palmas Altas, C/ Energía Solar 1, Ed. E, Seville (Spain). [email protected] JN: Departamento de Geometría y Topología. Facultad de Matemáticas, Universidad de Sevilla. C/ Tara s/n, 41012 Seville (Spain). jnv[email protected] 1
studied in several papers. For example, 1 -step solvable Lie algebras are the abelian ones. Abelian subalgebras and ideals are very useful in the study of Lie algebra contractions and degenerations. There is extensive literature on these topics, in particular for low-dimensional Lie algebras, as can be seen in [10, 19, 21] and the references given therein. Next, 2 -step solvable Lie algebras are called metabelian. These algebras have been studied in [1] by using the notion of weight graphs. Moreover, they admit abelian complex and Novikov structures, see [2, 9]. In [35] the author uses 3 -step solvable Lie algebras to construct a homogeneous conformally parallel Spin(7) metric on a certian solvmanifold. Finally, it is proved in [30] that 3 -step solvable Lie algebras contain the rst oscillator algebras with non-trivial semiequicontinuous coadjoint orbits. 2 Preliminaries In this section we recall some preliminary concepts, results and notations on Lie algebras. We have mainly followed [20, 25, 33, 36]. From here on, only nite-dimensional Lie algebras over the eld of complex numbers are considered. Given a Lie algebra g , a vector subspace h of g is an ideal if [h,g]⊆h . We denote by dim h the dimension of h as a vector space. The descendant central series of ideals, also known as the lower central series , of a given Lie algebra g is the ltration C1g⊇C2g⊇. . . ⊇Ckg⊇. . . where C1g=g and Ckg= [Ck−1g,g] for all k≥2. One has [Ckg, Cℓg]⊆Ck+ℓg for all k, ℓ ≥1 . Denition 1. A Lie algebra g is said to be nilpotent if there exists m∈N such that Cmg={0} . The smallest such m is called the nilpotency class (or the nilindex ) of g . The derived series of ideals of a given Lie algebra g is the ltration D0g⊇D1g⊇. . . ⊇Dkg⊇. . . where D0g=g and Dkg= [Dk−1g, Dk−1g] for all k≥1. Denition 2. A Lie algebra g is said to be solvable if there exists m∈N such that Dmg={0} . The smallest such m is called the derived length (or the solvability index , or even the solvindex ) of g . We say that g is m -step solvable if the derived length of g is m . The derived Lie algebra of g is by denition C2g=D1g= [g,g] . From here on, we will denote the derived algebra by Dg . A Lie algebra g satisfying Dg={0} is called abelian . If Dg is abelian, i.e. if D2g={0} , then g is said to be metabelian . 2
Let us notice that every nilpotent Lie algebra is solvable, since Dk−1g⊆Ckg for all k≥1 . Moreover, we have Dkg⊂C2kg , see e.g. [25, page 25]. This means in particular that if dim g≤ 2k , for some integer k , then the derived length of g is less than or equal to k . Denition 3. [37, Section 1.5] A n -dimensional Lie algebra g is said to be liform if its descendant central series satises dim(Ckg)=n−k for all 2≤k≤n. (1) Any n dimensional liform Lie algebra is nilpotent and its nilpotency class is n . A basis {e1, . . . , en} of a liform Lie algebra g is called adapted , see [37, Sec. 4.2], if the following relations 2 hold [e1, eh] = eh−1for 3 ≤h≤n, [e2, eh] = 0 for 1 ≤h≤n, [e3, eh] = 0 for 2 ≤h≤n. (2) As it was pointed out in [37, Section 4.2] any liform Lie algebra admits an adapted basis. In fact, with respect to an adapted basis, the ideals of the descendant central series are given by Ckg=⟨e2, . . . , en−k+1⟩ (3) where the angle brackets mean the C vector space generated by the corresponding vectors. A n -dimensional liform Lie algebra g is called a model liform Lie algebra (see [20]) if the only nonzero brackets in its law are [e1, eh] = eh−1 , for 3≤h≤n , where {e1, . . . , en} is an adapted basis of g . This condition is obviously independent of the chosen adapted basis in g . Notice that for n≤4 every liform Lie algebra is a model liform Lie algebra. 2.1 Two Numerical Invariants of Filiform Lie Algebras We recall the denition of two invariants of a liform Lie algebra g introduced in [16]. These invariants are dened for non-model liform Lie algebras, so we can assume that n > 4 . First, the invariant z1=z1(g) is dened as z1= max{k∈N|Cg(Cn−k+2g)⊇C2g} where Cg(h) is the centralizer of a given Lie subalgebra h of g , i.e. the set of elements in g whose bracket with any element of h is zero. The invariant z2=z2(g) is dened as z2= max {k∈N|Cn−k+1gis abelian} . Consequently, Cn−z2+1g is the largest abelian ideal in the lower central series of g . Remark 1. Notice that, for any n -dimensional non-model liform Lie algebra g , one has z2(g)≤ n−1 . Moreover, z2(g) = n−1 if and only if g is metabelian. 2 To simplify notations, the denition we give here is slightly dierent from, although equivalent to, the original one given in [37, Sec. 4.2]. 3
Equivalent denitions for the invariants z1 and z2 , more appropriate for practical use, are: z1= min {k≥4|[ek, en]= 0} and z2= min {k≥4|[ek, ek+1]= 0} , where {e1, . . . , en} is an adapted basis of g . These two invariants satisfy the following inequalities, see [16, Th. 15]: 4≤z1≤z2< n ≤2z2−2. (4) Given a n -dimensional non-model liform Lie algebra g , we use the triple (z1, z2, n) to summarize the information about both invariants and the dimension of g . 2.2 Previous results In [7, Lemme 1] Bratzlavsky obtained the general law for a liform metabelian Lie algebra, that is, a liform Lie algebra associated with the triple (z1, n −1, n) , for 4≤z1≤n−1 . In [26, Sec. 1] the author improves, for the innite dimensional case, the classication result of [7, Prop. 3] only valid in nite dimension. On the other hand, in [18, Lemma 1.2] the authors give the parametric expression of a list of basic brackets for any n -dimensional liform Lie algebra. Theorem 1. [7, Lemme 1] Let g be a liform Lie algebra of dimension n≥5 whose derived Lie algebra Dg is abelian. Then there exist a basis {x1, . . . , xn} of g and some complex numbers λ0, . . . , λn−5 such that [x1, xi] = xi+1 for 2 ≤i≤n−1; [xi, xj] = 0 for 3 ≤i<j≤nand [x2, xi] = n−i−2 X r=0 λrxi+2+rfor 3 ≤i≤n−2. Notice that when λ0=· · · =λn−5= 0 , we obtain the model liform Lie algebra since the equalities e1=x1, e2=xn, e3=xn−1, . . . , en=x2 dene an adapted basis of g . Notation 1. Given a n -dimensional Lie algebra g with basis {e1, . . . , en} and an integer 1≤ h≤n we denote by Ph:g→C the C -linear map dened as follows: for any vector u∈g , Ph(u) is the coordinate of u with respect to the basis vector eh, i.e. u= n X h=1 Ph(u)eh. The following theorem is stated in [13, Prop. 2] but we correct here a mistake there on the coecient of e2 in the brackets [ez1+k, ez2+ℓ] . More precisely, next theorem describes the law of any liform non-model Lie algebra, with associated triple (z1, z2, n) with respect to a suitable adapted basis. This result can be compared to the description of the ane variety of n dimensional liform Lie algebras given in [29, Sec. 4]. 4
Theorem 2. [13, Prop. 2] Let g be a n -dimensional non-model liform Lie algebra with associated triple (z1, z2, n) . Then, there exist an adapted basis {e1, . . . , en} of g and some complex numbers αi , γj and βkℓ , with 1≤i≤z2−z1+ 1 , 1≤j≤2n−z1−z2−2 , 2≤ℓ≤n−z2 , and 1≤k < z2−z1+ℓ , such that [e1, eh] = eh−1 for 3≤h≤n, [ez1+i, ez2+1] = α1ei+2 +α2ei+1 +· · · +αi+1e2 for 0≤i≤z2−z1, [ez1, ez2+j] = α1ej+1 +γ1ej+· · · +γj−1e2 for 2≤j≤n−z2, [ez1+k, ez2+ℓ] = k+ℓ X h=2 Ph([ez1+k−1, ez2+ℓ]+[ez1+k, ez2+ℓ−1]) eh+1 +βkℓ e2, for 2≤ℓ≤n−z2,1≤k < z2−z1+ℓ. Proof. The proof of this Theorem follows the one of [13, Prop. 2] bearing in mind that at the end of that proof, the coecient of e2 in the bracket [ez1+k, ez2+ℓ] is in general dierent from the coecients {αi} and {γj} . Remark 2. Notice rst that when all the parameters in Theorem 2 equal 0, i.e. αi=γj=βkℓ = 0 , we obtain the model liform Lie algebra of dimension n . Notice also that the Jacobi identity induces quadratic relations among the parameters αi , βkℓ and γj . These quadratic relations dene a certain non-empty Zariski closed set Z in the ane space Cµ of the parameters. Here the number of parameters is given by µ= (z2−z1+ 1) + (n−z2−1) + (n−z2−1)(n+z2−2z1) 2=(n−z2)(n+z2−2z1+ 1) 2. Moreover, the Lie algebras in Theorem 2 must be non-model and with numerical invariants (z1, z2, n) . These last conditions dene a certain Zariski open set of the previous closed set Z . Notation 2. The parametric family of liform Lie algebras described in Theorem 2 is denoted by Fαβγ . Notice that, by denition, the Lie algebras in this family have a xed associated triple (z1, z2, n) . In order to simplify notations, we drop this dependency in the notation Fαβγ . According to Theorem 2, the set of laws of n dimensional non-model liform Lie algebras is the disjoint union of the sets Fαβγ when (z1, z2, n) varies in the index set dened by inequalities (4); that is, when 4≤z1≤z2< n ≤2z2−2. Remark 3. We will see in the proof of Theorem 3 that if (z1, z2, n) satises 4≤z1≤2(n−z2)−4 and z1≤z2≤n−3≤2z2−5, then Fαβγ is the empty set, see Remark 5. 5
Let us notice that Theorem 2 implies Theorem 1 for non-model liform Lie algebras. Remark 4. According to Notation 1 and Theorem 2 the linear maps Ph satisfy the following relation Ph([ei, ej]) = Ph−1([ei−1, ej]+[ei, ej−1]) for 1 < i < j ≤n. This will be used in the next section. 3 Main results This section contains the main new results of this work. We state four Lemmas that are used to prove that the derived length of a family of nite dimensional liform Lie algebra is at most 3 (see Theorem 3). This family will be dened in Notation 5. We start with Lemma 1 in order to study the case where z2=n−2 . Lemma 1. Let g be a n -dimensional non-model liform Lie algebra with z2=n−2 . Then n≥6 and g has derived length 3 . Proof. First, the condition n≥6 follows from the fact that g is a non-model liform Lie algebra and from inequalities (4). Moreover, D2g is nonzero since z2=n−2 (see Remark 1). According to Theorem 2, we have the following general law for the algebra g , with respect to a suitable adapted basis [e1, eh] = eh−1,3≤h≤n, [ez1+i, en−1] = α1ei+2 +α2ei+1 +· · · +αi+1e2,0≤i≤n−z1−2, [ez1, en] = α1e3+γ1e2, [ez1+k, en] = k+2 X h=2 Ph([ez1+k−1, en]+[ez1+k, en−1])eh+1 +βk2e2,0< k < n −z1. According to equality (3), D2g⊆ ⟨e2, . . . , en−z1⟩ is abelian since n−z1< n −1 and then D3g={0} . Notation 3. We will use the following notations p= 2n−z1−2z2−3q=n−z2−2r=n−z1−1 Rk=r k−r k−1q kSk=r k−r k−1 q k−1. Given a real number y the expression ⌊y⌋ denotes the largest integer less than or equal to y . For the next three Lemmas, we will refer to the coecients αi , βkℓ , γj introduced in Theorem 2. If no confusion arises we use βk,ℓ instead of βkℓ . Moreover, in the proofs of Lemmas 2 and 3, we will use the following property that was proved in [17, Lemma 4]: [ei, ej] = 0 for 1 < i < z1, j > 1. 6
Lemma 2. [13, Prop. 6] Let g be a n -dimensional non-model liform Lie algebra. If the derived algebra Dg is not abelian, then α1= 0 . Proof. We give here an alternative proof to the one in [13, Prop. 6], which is related to other results of this section. We assume that g is a non-model liform Lie algebra with associated triple (z1, z2, n) . Let us consider the Jacobi identity J(ez1, en−1, en) = [[ez1, en−1], en] + [[en−1, en], ez1] + [[en, ez1], en−1] = 0. Notice rst that by Theorem 2, [ez1, en−1] = α1en−z2+γ1en−z2−1+· · · +γn−z2−2e2 and then we have that the rst term of the Jacobi identity is given by [[ez1, en−1], en] = α1[en−z2, en] + γ1[en−z2−1, en] + · · · +γn−z2−2[e2, en] = α1[en−z2, en] + γ1[en−z2−1, en] + · · · +γn−z2−z1[ez1, en]. Next, for the second term, we have [en−1, en] = q+r+2 X h=2 Ph([en−2, en])eh+1 +βn−1−z1,n−z2e2, so [ez1,[en−1, en]] = q+r+2 X h=2 Ph([en−2, en])[ez1, eh+1] + βn−1−z1,n−z2[ez1, e2] = q+r+2 X h=z1 Ph([en−2, en])[ez1, eh+1]. Finally, for the last term, we obtain [ez1, en] = α1en−z2+1 +γ1en−z2+· · · +γn−z2−1e2 [[ez1, en], en−1] = α1[en−z2+1, en−1] + γ1[en−z2, en−1] + · · · +γn−z2−1[e2, en−1]] = α1[en−z2+1, en−1] + γ1[en−z2, en−1] + · · · +γn−z1−z2+1[ez1, en−1]. Since Dg is not abelian, at least, one of the three terms of the Jacobi identity is not trivially null. In this way, J(ez1+m, en−1, en) = 0 is equivalent to α1[en−z2, en] + γ1[en−z2−1, en] + · · · +γn−z2−z1[ez1, en]− q+r+2 X h=z1 Ph([en−2, en])[ez1, eh+1] −α1[en−z2+1, en−1]−γ1[en−z2, en−1]− · · · − γn−z1−z2+1[ez1, en−1]=0. 7
The coecient of ep+4 in the Jacobi identity is given by α1Pp+3([eq+1, en]+[eq+2, en−1]) −Pq+r+2([en−2, en])− Pp+3([eq+2, en−1]+[eq+1, en−2])= α1Pp+4([eq+2, en]) −Pq+r+2([en−2, en]) −Pp+4([eq+3, en−1]). Now, from the recursive denition of the linear maps Ph (see Remark 4) and Vandermonde's identity for combinatorial numbers, we obtain Pn−z2+1([ez1+k, en−k]) = q+ 1 kα1. Consequently, Pp+4([eq+2, en]) = p+ 2 r−z2+ 1α1 Pq+r+2([en−2, en]) = q+r r−1α1 Pp+4([eq+3, en−1]) = p+ 2 r−z2+ 2α1. We conclude that the coecient of ep+4 in the Jacobi identity is given by α2 1 p+ 2 r−z2+ 1−q+r r−1−p+ 2 r−z2+ 2= α2 1−z1−2 q+ 1 p+ 2 q−q+r r−1. The previous expression is zero if and only if α1= 0 . Notation 4. We recall here the expressions introduced in Notation 3 and introduce some more useful notations am=m+q−1 mm+p q−m+q mm+p q−1−p+m m ⌊r 2⌋ X k=0 Rk, bm=m+q−1 mm+p q+1 −m+q−1 m−1m+p q−m+q mm+p q −m+q m−1m+p q−1−m+p m−1P⌊r 2⌋ k=0 Rk−m+p m ⌊r 2⌋ X k=0 Sk, cm=m+q−1 m−1m+p q+1 −m+q m−1m+p q−m+p m−1 ⌊r 2⌋ X k=0 Sk. 8
Lemma 3. Let g be a n -dimensional non-model liform Lie algebra associated with the triple (z1, z2, n) , whose derived algebra Dg is not abelian. Then, for m= 0, . . . , r −1 , the coecient of the vector em+p+2 in the Jacobi identity J(ez1+m, en−1, en) = 0 is amγ2 1+bmγ1α2+cmα2 2. Proof. First, from Lemma 2, we can suppose that α1= 0 in Theorem 2. Next, by using that theorem, Remark 4 and Vandermonde's identity for combinatorial numbers, we obtain the following relation Pn−z2([ez1+k, en−k]) = q kγ1+q k−1α2. (5) Now, let us consider the Jacobi identity J(ez1+m, en−1, en) = [[ez1+m, en−1], en] + [[en−1, en], ez1+m] + [[en, ez1+m], en−1] = 0. Notice rst that from Theorem 2 we have [ez1+m, en−1] = m+q X h=2 Ph([ez1+m−1, en−1]+[ez1+m, en−2])eh+1 +βm,n−z2−1e2 and then [[ez1+m, en−1], en] = m+q X h=2 Ph([ez1+m−1, en−1]+[ez1+m, en−2])[eh+1, en] + βm,n−z2−1[e2, en] = m+q X h=z1−1 Ph([ez1+m−1, en−1]+[ez1+m, en−2])[eh+1, en]. Next, for the second term, we have [en−1, en] = q+r+1 X h=2 Ph([en−2, en])eh+1 +βn−z1−1,n−z2e2, so [ez1+m,[en−1, en]] = q+r+1 X h=2 Ph([en−2, en])[ez1+m, eh+1] + βn−z1−1,n−z2[ez1+m, e2] = q+r+1 X h=z1 Ph([en−2, en])[ez1+m, eh+1]. Finally, for the last term, we obtain [ez1+m, en] = m+q+1 X h=2 Ph([ez1+m−1, en]+[ez1+m, en−1])eh+1 +βm,n−z2e2 [[ez1+m, en], en−1] = m+q+1 X h=2 Ph([ez1+m−1, en]+[ez1+m, en−1])[eh+1, en−1] + βm,n−z2[e2, en−1] 9
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