Abstract: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 $r$.
| Comments: | 21 pages; comments are welcome |
| Subjects: | Representation Theory (math.RT); K-Theory and Homology (math.KT); Rings and Algebras (math.RA) |
| MSC classes: | 17A30, 17A36, 17A40, 17B40 |
| Cite as: | arXiv:2605.25576 [math.RT] |
| (or arXiv:2605.25576v1 [math.RT] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25576 arXiv-issued DOI via DataCite (pending registration) |
From: Apurba Das [view email]
[v1]
Mon, 25 May 2026 08:29:22 UTC (25 KB)
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