The nilpotent variety of W(1;n)p is irreducible

Abstract

In the late 1980s, Premet conjectured that the nilpotent variety of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic p > 0 is irreducible. This conjecture remains open, but it is known to hold for a large class of simple restricted Lie algebras, e.g. for Lie algebras of connected reductive algebraic groups, and for Cartan series W, S and H. In this paper, with the assumption that p > 3, we confirm this conjecture for the minimal p-envelope W (1; n) p of the Zassenhaus algebra W (1; n) for all n ≥ 2.