Elementary subalgebras of Lie algebras

TL;DR

This paper explores the geometric structure of the variety of elementary subalgebras of a Lie algebra and how it relates to the representation theory of the algebra, revealing new connections between geometry and algebraic representations.

Contribution

It introduces the variety $E(r,g)$ of elementary subalgebras and demonstrates its significance in understanding Lie algebra representations and associated vector bundles.

Findings

Identified geometric structures of $E(r,g)$ for various Lie algebras.

Linked algebraic vector bundles on $E(r,g)$ to specific classes of representations.

Connected the geometry of $E(r,g)$ with the representation theory of algebraic groups.

Abstract

We initiate the investigation of the projective variety $E(r,g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $g$ for some $r > 0$ and demonstrate that this variety encodes considerable information about the representations of $g$. For various choices of $g$ and $r$, we identify the geometric structure of $E(r,g)$. We show that special classes of (restricted) representations of $g$ lead to algebraic vector bundles on $E(r,g)$. For $g = Lie(G)$ the Lie algebra of an algebraic group $G$, rational representations of $G$ enable us to realize familiar algebraic vector bundles on $G$-orbits of $E(r, g)$.