Cohomology and deformations of Relative Rota-Baxter operators on Lie-Yamaguti algebras

TL;DR

This paper develops a cohomology theory for relative Rota-Baxter operators on Lie-Yamaguti algebras and uses it to analyze their deformations, linking cohomology classes to deformation equivalence and extension obstructions.

Contribution

It introduces a cohomology framework for these operators and applies it to characterize deformations and their extensions in Lie-Yamaguti algebras.

Findings

Cohomology classifies equivalent deformations.

Obstruction classes determine extendability of deformations.

First cohomology relates to infinitesimal deformations.

Abstract

In this paper, we establish the cohomology of relative Rota-Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then we use this type of cohomology to characterize deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota-Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order $n$ deformation of a relative Rota-Baxter operator can be extended to an order $n+1$ deformation if and only if the obstruction class in the second cohomology group is trivial.