Vector Bundles Associated to Lie Algebras

TL;DR

This paper develops a functorial method to associate coherent sheaves and algebraic vector bundles to representations of restricted Lie algebras, linking algebraic and geometric structures on varieties of elementary subalgebras.

Contribution

It introduces a new construction connecting restricted Lie algebra representations to sheaves and vector bundles on subvarieties of the elementary subalgebra variety.

Findings

Representations of constant radical or socle rank yield algebraic vector bundles.

The construction applies to Lie algebras of algebraic groups, relating to familiar vector bundles.

The approach generalizes modules of constant Jordan type to geometric objects.

Abstract

We introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra $\mathfrak g$. These are sheaves on locally closed subvarieties of the projective variety $\mathbb E(r,\mathfrak g)$ of elementary subalgebras of $\mathfrak g$ of dimension $r$. We show that representations of constant radical or socle rank studied in \cite{CFP3} which generalize modules of constant Jordan type lead to algebraic vector bundles on $\mathbb E(r,\mathfrak g)$. For $\mathfrak g = Lie(G)$, the Lie algebra of an algebraic group $G$, rational representations of $G$ enable us to realize familiar algebraic vector bundles on $G$-orbits of $\mathbb E(r, \mathfrak g)$.