Representations of Leibniz Algebras

TL;DR

This paper explores the structure of representations of Leibniz algebras, showing how they relate to Lie algebra representations and highlighting differences in decomposability for semisimple cases.

Contribution

It establishes a method to derive irreducible Leibniz algebra representations from Lie algebra representations and clarifies the limitations of decomposing these representations.

Findings

Irreducible Leibniz representations derive from semisimple Lie algebra representations

Decomposition into irreducibles does not always hold for semisimple Leibniz algebras

Provides insights into the structure of Leibniz algebra representations

Abstract

In this paper we prove that every irreducible representation of a Leibniz algebra can be obtained from irreducible representations of the semisimple Lie algebra from the Levi decomposition. We also prove that - in general - for (semi)simple Leibniz algebras it is not true that a representation can be decomposed to a direct sum of irreducible components.