A note on representations of Lie-Yamaguti algebras induced by left Leibniz algebras
A. Nourou Issa
TL;DR
This note shows that every left representation of a left Leibniz algebra naturally induces a representation of the associated Lie-Yamaguti algebra, with explicit formulae $\rho(x) = l(x) - r(x)$, $\theta(x,y) = - r(y)r(x)$, and $D(x,y) = - l(x\cdot y)$, and verifies the LY axioms (R1)-(R7). It also proves that equivalence of Leibniz representations implies equivalence of the induced LY representations, establishing a compatibility between the two representation theories. The results are illustrated via examples, including the adjoint representation, dual representations, and special cases where $r=0$ or $r=-l$, clarifying how LY representations arise from familiar Leibniz structures. This provides a concrete bridge from Leibniz representation theory to LY representation theory, enabling construction of LY modules from known Leibniz modules.
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 $(\mathfrak{g}, [,], [\![ , , ]\!])$.