Locally Hopf varieties of Lie algebras

Translated from Matematicheskie Zametki, Vol. 45, No. 6, pp. 56–60, June, 1989.     

Nilpotent commuting varieties of reductive Lie algebras

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p0, and =LieG. In positive characteristic, suppose in addition that p is good for G and the derived subgroup of G is simply connected. Let =() denote the nilpotent variety of , and ^nil():={(x,y)×|[x,y]=0}, the nilpotent commuting variety of . Our main goal in this paper is to show that the variety ^nil() is equidimensional. In characteristic 0, this confirms a conjecture of Vladimir Baranovsky; see [2]. When applied to GL(n), our result in conjunction with an observation in [2] shows that the punctual (local) Hilbert scheme _n Hilb ⁿ (²) is irreducible over any algebraically closed field. Mathematics Subject Classification (2000) 20G05     

Two remarks on varieties of Lie algebras

The varieties of Lie algebras in which every subalgebra of the free algebra is also free are completely described, and a proof is given for a theorem on the verbal ideals of the ideals of an absolutely free Lie algebra.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 389–398, October, 1968.The author takes this opportunity to thank A. L. Shmel'kin for suggesting the proof of the analog of the Neumann-Weigold theorem for varieties of Lie algebras.     

Almost locally representable varieties of Lie algebras

Translated from Matematicheskie Zametki, Vol. 51, No. 3, pp. 9–15, March, 1992.     

On varieties parametrizing graded complex Lie algebras

We study spaces parametrizing graded complex Lie algebras from geometric as well as algebraic point of view. If R is a finite-dimensional complex Lie algebra, which is graded by a finite abelian group of order n, then a graded contraction of R, denoted by , is defined by a complex n × n-matrix , i, j = 1, . . . , n. In order for to be a Lie algebra, should satisfy certain homogeneous equations. In turn, these equations determine a projective variety X _R . We compute the first homology group of an irreducible component M of X _R , under some assumptions on M. We look into algebraic properties of graded Lie algebras where .      

The polynomial growth of colength of varieties of Lie algebras

It is proved that the colength of every API-variety of Lie algebras grows polynomially, and we give a number of examples in which the colength grows more rapidly than any polynomial function does. These indicate that for many of the important varieties of Lie algebras, such as varieties of solvable algebras of derived length 3, varieties generated by some infinite-dimensional simple algebras of Cartan type, or by certain Katz-Mudi algebras, the growth of colength will be superpolynomial. Supported by RFFR grants No. 96-01-00146 and No. 96-15-96050. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 161–175, March–April, 1999.     

Automorphic equivalence in the varieties of representations of Lie algebras

Abstract

In this paper, we consider the wide class of subvarieties of the variety of all representation of Lie algebras over a field k of characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations from these subvarieties.