A note on representations of Lie-Yamaguti algebras induced by left Leibniz algebras

TL;DR

This paper explores how representations of left Leibniz algebras naturally induce representations of associated Lie-Yamaguti algebras, establishing a correspondence between their representation theories.

Contribution

It demonstrates that every left representation of a left Leibniz algebra induces a corresponding representation of the associated Lie-Yamaguti algebra, and shows the equivalence of such representations.

Findings

Induction of Lie-Yamaguti representations from Leibniz algebra representations

Equivalence of Leibniz algebra representations implies equivalence of induced Lie-Yamaguti representations

Provides a structural link between Leibniz and Lie-Yamaguti algebra representations

Abstract

It is well-known that each left Leibniz algebra has a natural structure of a Lie-Yamaguti algebra. In this paper it is shown that every left representation of a left Leibniz algebra $(\mathfrak{g}, \cdot)$ induces naturally a representation of the Lie-Yamaguti algebra $(\mathfrak{g}, [,], [\![ , , ]\!])$ that is associated with $(\mathfrak{g}, \cdot)$. Moreover, it is proved that equivalent representations of $(\mathfrak{g}, \cdot)$ give equivalent representations of $(\mathfrak{g}, [,], [\![ , , ]\!])$.