Blocks of representations of Lie algebras

TL;DR

This paper extends the concept of block decomposition from finite group representations to finite-dimensional Lie algebras, providing a natural classification of their irreducible modules, especially for supersoluble algebras.

Contribution

It introduces a classification of irreducible Lie algebra representations into blocks, analogous to group theory, and explores this for supersoluble algebras.

Findings

Defined a block classification for Lie algebra modules

Established the finest natural decomposition of L-modules

Analyzed block structure for supersoluble algebras

Abstract

In the theory of finite groups, the irreducible representations of G over a field F are classified into blocks based on a direct decompositions of the group algebra FG. This gives a natural decomposition of FG-modules into direct summands, each summand having all its composition factors belonging to a single block. This block decomposition is the finest natural decomposition of the FG-modules. In this paper, a classification of the irreducible representations of a finite dimensional Lie algebra L into blocks is defined, giving the finest natural direct decomposition of L-modules. This classification is investigated for supersoluble algebras.