Unimodal betti numbers for a class of nilpotent lie algebras

In this paper we prove that all finite dimens~onal complex nilpotent Lie algebras containing an abelian ideal of codimension 1 have unirnodal Betti numbers.     

Identities of representations of nilpotent lie algebras

Lret N be a nilpotent of class 2 Lie algebra with one-dimensional centre C = Kc over an infinite field K and let p : N → End_k:(V) be a representation of N in a vector space V such that p(c) is invertible in End_k(V). We find a basis for the identities of the representation p. As consequences we obtain a basis for all the weak polynomial identities of the pair (M₂:(K), s1₂(K)) over an infinite field K of characteristic 2 and describe the identities of the regular representation of Lie algebras related with the Weyl algebra and its tensor powers.     

Classes of 2-step nilpotent lie algebras

We give a characterization of the Lie algebras of H-type independent of the inner product used in the definition. We classify the real 2-step nilpotent Lie algebras with 2-dimensional center. Using these results we give examples of regular Lie algebras that are not H-type.     

Ideals of identities of representations of nilpotent lie algebras

Let L be a Lie algebra, nilpotent of class 2, over an infinite field K, and suppose that the centre C of L is one dimensional; such Lie algebras are called Heisenberg algebras. Let ρ:L→hom _KV be a finite dimensional representation of the Heisenberg algebra L such that ρ(C) contains non-singular linear transformations of V, and denote l(ρ) the ideal of identities for the representation ρ. We prove that the ideals of identities of representations containing I(ρ) and generated by multilinear polynomials satisfy the ACC. Let sl ₂(L) be the Lie algebra of the traceless 2×2 matrices over K, and suppose the characteristic of K equals 2. As a corollary we obtain that the ideals of identities of representations of Lie algebras containing that of the regular representation of sl ₂(K) and generated by multilinear polynomials, are finitely based. In addition we show that one cannot simply dispense with the condition of multilinearity. Namely, we show that the ACC is violated for the ideals of representations of Lie algebras (over an infinite field of characteristic 2) that contain the identities of the regular representation of sl ₂(K).     

Exponents of varieties of lie algebras with a nilpotent commutator subalgebra

We prove that the growth exponent of any variety of Lie algebras with a nilpotent commutator subalgebra is integral.     

On the Ricci curvature of solvable metric lie algebras with two-step nilpotent derived algebras

We prove that the Ricci operator of any nonunimoular solvable metric Lie algebra having a two-step nilpotent derived Lie algebra of dimension 6 has at least two negative eigenvalues.     

On the dimension of the Schur multiplier of nilpotent lie algebras

We give a bound on the dimension of the Schur multiplier of a finite dimensional nilpotent Lie algebra which sharpens the earlier known bounds.     

Lower bounds for algebraic algorithms for nilpotent and solvable lie algebras

We obtain the lower bounds for the tensor rank for the class of nilpotent and solvable Lie algebras (in terms of dimensions of certain quotient algebras). These estimates, in turn, give lower bounds for the complexity of algebraic algorithms for this class of algebras. We adduce examples of attainable estimates for nilpotent Lie algebras of various dimensions.     

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ?-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.     

On characterizing nilpotent lie algebras by their multiplierst(L) = 3,4,5,6

The nilpotent Lie algebras L of dimension n whose multipliers have dimension ½n(n-1)-t(L) have been found in [2] for t(L) = 0,1,2. Using a different method, we find similar results for t(L) = 3,4,5,6. The first author is extending the results to t(L) = 7 and 8.