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

TL;DR

This paper develops a cohomology framework for relative Rota-Baxter operators on Lie-Yamaguti algebras, using $L_ abla$-algebra structures and Maurer-Cartan elements to understand their deformations.

Contribution

It constructs a differential graded Lie algebra for Lie-Yamaguti algebra deformations and characterizes Rota-Baxter operators as Maurer-Cartan elements within an $L_ abla$-algebra.

Findings

Established cohomology theory for Rota-Baxter operators on Lie-Yamaguti algebras.

Linked twisted $L_ abla$-algebra structures with cohomology.

Classified certain algebra deformations using the developed cohomology.

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.