Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras

TL;DR

This paper characterizes the representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras, linking them to representations of the enveloping algebra with added structure, and extends this to infinitesimal s-manifolds.

Contribution

It provides a description of Lie-Yamaguti algebra representations in terms of their semisimple enveloping Lie algebras, including additional data, and extends to infinitesimal s-manifolds.

Findings

Representations of Lie-Yamaguti algebras are described via their semisimple enveloping Lie algebras.

Any representation of an infinitesimal s-manifold with semisimple enveloping algebra derives from a Lie algebra representation.

Abstract

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