Sandwich elements and the Richardson property

TL;DR

This paper characterizes restricted Lie algebras with irreducible Richardson modules, showing they are isomorphic to Lie algebras of reductive algebraic groups, thus linking module properties to algebraic group structures.

Contribution

It establishes a classification result connecting irreducible Richardson modules to reductive Lie algebras over algebraically closed fields.

Findings

Irreducible Richardson modules imply the Lie algebra is reductive.

The structure of the module determines the algebra's isomorphism class.

The result applies to restricted Lie algebras over fields with characteristic p > 3.

Abstract

Let $\mathcal{L}$ be finite dimensional restricted Lie algebra over an algebraically closed field $k$ of characteritic $p>3$. A finite dimensional restricted $\mathcal{L}$-module $V$ is called Richardson if $V$ is faithful and there exists a subspace $R$ of $\mathfrak{gl}(V)$ such that $[\mathcal{L},R]\subseteq R$ and $\mathfrak{gl}(V)=\mathcal{L}\oplus R$, where we identify $\mathcal{L}$ with its image in $\mathfrak{gl}(V)$. In this paper we show that if $\mathcal{L}$ admits an irreducible Richardson module then it is isomorphic (as a restricted Lie algebra) to the Lie algebra of a reductive $k$-group.