Wide subalgebras of semisimple Lie algebras

TL;DR

This paper investigates large classes of wide subalgebras in semisimple Lie algebras, characterizing their properties and relationships with indecomposable modules and epimorphic subgroups.

Contribution

It introduces a systematic study of wide subalgebras, describing their structure and connection to invariant endomorphisms and epimorphic subgroups.

Findings

Identifies several classes of wide subalgebras

Establishes a link between wide subalgebras and indecomposable modules

Explores the relationship with epimorphic subgroups

Abstract

Let G be a connected semisimple algebraic group over $k$, with Lie algebra $\g$. Let $\h$ be a subalgebra of $\g$. A simple finite-dimensional $\g$-module V is said to be $\h$-indecomposable if it cannot be written as a direct sum of two proper $\h$-submodules. We say that $\h$ is wide, if all simple finite-dimensional $\g$-modules are $\h$-indecomposable. Some very special examples of indecomposable modules and wide subalgebras appear recently in the literature. In this paper, we describe several large classes of wide subalgebras of $\g$ and initiate their systematic study. Our approach is based on the study of idempotents in the associative algebra of $\h$-invariant endomorphisms of V. We also discuss a relationship between wide subalgebras and epimorphic subgroups.