Invariant Nonassociative Algebra Structures on Irreducible Representations of Simple Lie Algebras

Abstract

An irreducible representation of a simple Lie algebra can be a direct summand of its own tensor square. In this case, the representation admits a nonassociative algebra structure which is invariant in the sense that the Lie algebra acts as derivations. We study this situation for the Lie algebra {\small $sl(2)$}.