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

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 $(\mathfrak{g},[,],[[,,]])$.