Disjointness of a simple matrix Lie group and its Lie algebra

TL;DR

This paper investigates the intersection properties of simple Lie groups and their Lie algebras within the general linear group, establishing conditions under which they intersect based on classical group classification and minuscule representations.

Contribution

It characterizes when a simple Lie group and its Lie algebra intersect as subsets of matrices, linking this to classical groups and minuscule representations.

Findings

Intersection occurs only for classical groups.

Minuscule representations are key to the intersection.

Provides a criterion for disjointness of groups and algebras.

Abstract

Let $G$ be a connected closed subgroup of $\mathrm{GL}_n(\mathbb{C})$ which is simple as a Lie group and which acts irreducibly on $\mathbb{C}^n$. Regarding both $G$ and its Lie algebra $\mathfrak{g}$ as subsets of $M_n(\mathbb{C})$, we have $G\cap \mathfrak{g}\neq\emptyset$ if and only if $G$ is a classical group and $\mathbb{C}^n$ is a minuscule representation.