The nilpotent variety of $W(1;n)_{p}$ is irreducible

TL;DR

This paper proves that the nilpotent variety of the minimal p-envelope of the Zassenhaus algebra W(1;n) is irreducible for all n ≥ 2, confirming a special case of Premet's conjecture in characteristic p > 3.

Contribution

It establishes the irreducibility of the nilpotent variety for the minimal p-envelope of W(1;n), a previously unresolved case in the theory of restricted Lie algebras.

Findings

Nilpotent variety of W(1;n)_p is irreducible for all n ≥ 2

Confirms Premet's conjecture for a new class of Lie algebras

Valid for characteristic p > 3

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 $n\geq 2$.