Filiform Lie Algebras with Low Derived Length
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Filiform Lie Algebras with Low Derived Length un ˜ez-Vald´es F. J. Castro-Jim´enez , M. Ceballos and J. N´ Abstract. We construct, for any n ≥ 5, a family of complex filiform Lie algebras with derived length at most 3 and dimension n. We also give examples of n-dimensional filiform Lie algebras with derived length greater than 3. Mathematics Subject Classification. 17B30, 17-08, 17B05, 68W30. Keywords. Filiform Lie algebra, derived length, Lie algebra invariants.
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 finite p-groups this study was initiated by Burnside [15,16] and then continued by several authors (e.g. Hall, Magnus, Itˆ o, Higman, Blackburn, Mann 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 [19], Patterson [33,34], Bokut [9]. In [13] the authors show that there are filiform Lie algebras of arbitrary derived length. They describe, for any k ≥ 2, a filiform Lie algebra of derived length k and dimension n for each n satisfying 2k ≤ n + 1 < 2k+1 . These algebras were studied by Benoist [8]. Filiform Lie algebras were introduced by Vergne [41] and they have been classified up to dimension 8; see [2] and a corrected version of this classification given in [36]. Moreover, nilpotent Lie algebras are classified up to dimension 7 in [24]. In view of the classification of filiform Lie algebras, several invariants have been introduced in the literature. For example, in [20] two numerical invariants were introduced and studied. Our results on the derived length of filiform Lie algebras are based on these invariants. Recently, some new invariants have been defined in order to study these algebras. In particular, the notion of breath is introduced in [29] and a generalization of it, by the so-called characteristic sequence, in [35]. In addition, Lie algebras graded by an abelian group are analyzed in [3]. The special case of graded filiform Lie algebras is studied in [4–6]. 0123456789().: V,-vol
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In this paper we construct, for any n ≥ 5, a family of complex filiform Lie algebras with derived length at most 3 and dimension n, see Theorem 3. To this end we improve the result [17, Prop. 2] and describe the law of any filiform 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 affine variety of n-dimensional filiform Lie algebras given in [31, Sec. 4], where the variety of Lie algebras of maximal class is described. See also [38] where the related notion of narrow Lie algebras is treated. Theorem 2 generalizes a result of Bratzlavsky [10] valid for metabelian Lie algebras. We also construct a family of complex filiform Lie algebras of dimension 15 and derived length 4 generalising the one given in [11, Ex. 3.2]. Let us rem
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