Casimir operators, abelian subspaces and u-cohomology

TL;DR

This survey explores the relationships between the exterior algebra of simple Lie algebras, the Casimir operator, and u-cohomology, summarizing both historical and recent results in the field.

Contribution

It provides a self-contained exposition of Kostant's results connecting Casimir operators, abelian subspaces, and u-cohomology, highlighting recent developments.

Findings

Clarifies the role of Casimir operators in Lie algebra cohomology

Summarizes key results of Kostant and colleagues

Provides a comprehensive overview of the algebraic structures involved

Abstract

This survey paper is an exposition of old and recent results of Kostant and al. on the relationships between the exterior algebra of a simple Lie algebra and the action of the Casimir operator on it. Our exposition relies on u-cohomology and it is basically self-contained.