Maurer-Cartan characterization, $L_{\infty }$-algebras, and cohomology of relative Rota-Baxter operators on Lie-Yamaguti algebras

Abstract

In this paper, we first construct a differential graded Lie algebra that controls deformations of a Lie-Yamaguti algebra. Furthermore, a relative Rota-Baxter operator on a Lie-Yamaguti algebra is characterized as a Maurer-Cartan element in an appropriate $L_{\infty }$-algebra that we build through the graded Lie bracket of Lie-Yamaguti algebra's controlling algebra, and gives rise to a twisted $L_{\infty }$-algebra that controls its deformation. Next we establish the cohomology theory of relative Rota-Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then we clarify the relationship between the twisted $L_{\infty }$-algebra and the cohomology theory. Finally as byproducts, we classify certain deformations on Lie-Yamaguti algebras using the cohomology theory.

This paper surpercedes the preprint arXiv: 2204.04872. We only consider cohomology and deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras in the preprint arXiv: 2204.04872. Not only do we consider cohomology and deformations but also controlling algebras of Lie-Yamaguti algebras and their relative Rota-Baxter operators are constructed in the present paper