Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras

Abstract

Let T be a Lie-Yamaguti algebra whose standard enveloping Lie algebra $L\left(T\right)$ is semisimple and $[T,T,T]=T$. Then we give a description of representations of T in terms of representations of $L\left(T\right)$ with certain additional data. Similarly, if $(T,\sigma )$ is an infinitesimal s-manifold such that $L\left(T\right)$ is semisimple, then any representation of $(T,\sigma )$ comes from a representation of $L\left(T\right)$.