Representations and Cocycle Twists of Color Lie Algebras

TL;DR

This paper explores the relationship between finite-dimensional representations of color Lie algebras and their cocycle twists, demonstrating that cocycle twists preserve the FCR-property and explicitly classifying representations of a specific algebra.

Contribution

It establishes that cocycle twists maintain the FCR-property and provides a complete classification of finite-dimensional representations of the color Lie algebra sl_2^c.

Findings

Cocycle twists preserve the FCR-property.

Complete classification of finite-dimensional representations of sl_2^c.

Universal enveloping algebras are key tools in the analysis.

Abstract

We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra $sl_2^c$.