Factorizations, classifying complements problem and deformation maps for Lie-Yamaguti algebras

A Lie-Yamaguti algebra is a non-associative algebraic structure that generalizes both Lie algebras and Lie triple systems. We first consider the factorization problem for Lie-Yamaguti algebras that essentially related to the bicrossed product of Lie-Yamaguti algebras. Next, given an inclusion $\mathfrak{g} \subset E$ of Lie-Yamaguti algebras and a strong $\mathfrak{g}$-complement $\mathfrak{h}$, we describe and classify all $\mathfrak{g}$-complements in $E$. In particular, we show that any other $\mathfrak{g}$-complement in $E$ is isomorphic to $\mathfrak{h}$ by some deformation map $r: \mathfrak{h} \rightarrow \mathfrak{g}$. Despite this importance, it turns out that a deformation map generalizes homomorphisms, derivations, crossed homomorphisms and relative Rota-Baxter operators on Lie-Yamaguti algebras. We define the cohomology of a deformation map unifying the cohomologies of all the operators mentioned above. Finally, we provide a Maurer-Cartan characterization and construct the governing $L_\infty$-algebra of a deformation map $r$ that controls the linear deformations of $r$.