On finite W-algebras for Lie algebras and superalgebras

Abstract

The finiteW-algebras are certain associative algebras associated to a complex semi-simple or reductive Lie algebra g and a nilpotent element e of g. Due to recent results of I. Losev, A. Premet and others, finite W-algebras play a very important role in description of primitive ideals. In the full generality, the finiteW-algebras were introduced by A. Premet. It is a result of B. Kostant that for a regular nilpotent (principal) element e, the finiteW-algebra coincides with the center of U(g). Premet’s definition makes sense for a simple Lie superalgebra g = g¯0 g¯1 in the case when g¯0 is reductive, g admits an invariant super-symmetric bilinear form, and e is an even nilpotent element. We show that certain results of A. Premet can be generalized for classical Lie superalgebras. We consider the case when e is an even regular nilpotent element. The associated finiteW-algebra is called principal. Kostant’s result does not hold in this case. This is joint work with V. Serganova.