On representations of Lie algebras compatible with a grading

TL;DR

This paper explores how Lie algebra representations can be compatible with gradings, focusing on finite-dimensional representations of sl(n,C) and utilizing the Gel'fand-Tseitlin method for effective analysis.

Contribution

It introduces the concept of grading-compatible representations and applies it specifically to sl(n,C), extending existing Lie algebra representation theory.

Findings

Effective use of Gel'fand-Tseitlin method for graded representations

Characterization of representations compatible with Z2-gradings

Extension of representation theory to graded Lie algebras

Abstract

The paper extends existing Lie algebra representation theory related to Lie algebra gradings. The notion of a representation compatible with a given grading is defined and applied to finite-dimensional representations of $sl(n,\mathbb{C})$ in relation with its $\mathbb{Z}_2$-gradings. For representation theory of $sl(n,\mathbb{C})$ the Gel'fand-Tseitlin method turned out very effective.