Nilpotent commuting varieties of the Witt algebra

TL;DR

This paper investigates the structure of the nilpotent commuting variety of the Witt algebra over an algebraically closed field, showing it is reducible and describing its components and dimensions, extending classical Lie algebra results.

Contribution

It extends Premet's results to the Witt algebra, demonstrating the reducibility and equidimensionality of its nilpotent commuting variety and explicitly characterizing its components.

Findings

The nilpotent commuting variety of the Witt algebra is reducible.

The variety is equidimensional with explicitly determined components.

Nilpotent commuting varieties of Borel subalgebras are also characterized.

Abstract

Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathscr{N}={x\in\ggg\mid x^{[p]}=0}$ be the nilpotent variety of $\mathfrak{g}$, and $\mathscr{C}(\mathscr{N}):=\{(x,y)\in \mathscr{N}\times\mathscr{N}\mid [x,y]=0\}$ the nilpotent commuting variety of $\mathfrak{g}$. As an analogue of Premet's result in the case of classical Lie algebras [A. Premet, Nilpotent commuting varieties of reductive Lie algebras. Invent. Math., 154, 653-683, 2003.], we show that the variety $\mathscr{C}(\mathscr{N})$ is reducible and equidimensional. Irreducible components of $\mathscr{C}(\mathscr{N})$ and their dimension are precisely given. Furthermore, the nilpotent commuting varieties of Borel subalgebras are also determined.