Elementary Subalgebrs of Lie Algebras

TL;DR

This paper explores the geometric structure of varieties of elementary subalgebras in Lie algebras, introducing invariants that relate to support varieties and analyzing their properties and examples.

Contribution

It initiates the study of these varieties for various ranks and introduces new invariants, expanding understanding of their geometric and representation-theoretic significance.

Findings

Identification of special cases with interesting geometric structures

Introduction of local radical and socle rank invariants

Examples of modules with constant rank functions

Abstract

We initiate the investigation of the projective varieties $\mathbb E(r,\mathfrak g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $\mathfrak g$ for various $r \geq 1$. These varieties $\mathbb E(r,\mathfrak g)$ are the natural ambient varieties for generalized support varieties for restricted representations of $\mathfrak g$. We identify these varieties in special cases, revealing their interesting and varied geometric structures. We also introduce invariants for a finite dimensional $\mathfrak u(\mathfrak g)$-module $M$, the local $(r,j)$-radical rank and local $(r,j)$-socle rank, functions which are lower/upper semicontinuous on $\mathbb E(r,\mathfrak g)$. Examples are given of $\mathfrak u(\mathfrak g)$-modules for which some of these rank functions are constant.