Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type A.

An improvement of a bound of Yankosky (2003) is presented in this paper, thanks to a restriction which has been recently obtained by the authors on the Schur multiplier M(L) of a finite dimensional nilpotent Lie algebra L. It is also described the structure of all nilpotent Lie algebras such that the bound is attained. An important role is played by the presence of a derived subalgebra of maxim...

We consider the construction of a nilpotent BRST charge for extensions of the Virasoro algebra of the form {Ta, Tb} =f,{Tc+ V~,~TcTd, (classical algebras in terms of Poisson brackets) and [To, TbJ = habI + f~{ T~ + V~ (T~ Te) (quantum algebras in terms of commutator brackets; normal ordering of the product (T~ Te) is understood). In both cases we assume that the set of generators {To} splits in...

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue...

We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova.

We investigate the properties of Lie algebras of characteristic p which satisfy the Engel identity xy" = 0 for some n < p. We establish a criterion which (when satisfied) implies that if a and b are elements of an Engel-n Lie algebra L then abn~2 generates a nilpotent ideal of I. We show that this criterion is satisfied for n = 6, p = 1, and we deduce that if G is a finite m-generator group of ...

This is the first of a series of papers devoted to certain pairs of commuting nilpotent elements in a semisimple Lie algebra that enjoy quite remarkable properties and which are expected to play a major role in Representation theory. The properties of these pairs and their role is similar to those of the principal nilpotents. To any principal nilpotent pair we associate a two-parameter analogue...

The ground field k is algebraically closed and of characteristic zero. Throughout, G is a connected and simply connected semisimple algebraic group, g is its Lie algebra, and Φ is the Cartan–Killing form on g; l = rk g. Int g (resp. Aut g) is the group of inner (resp. all) automorphisms of g; N is the nilpotent cone in g. For x ∈ g, z(x) is the centraliser of x in g. Let g = ⊕i∈Zmgi be a period...