Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs

TL;DR

This paper develops a cohomology theory for LieYRep pairs, introduces relative Rota-Baxter-Nijenhuis structures, and explores their properties and relationships with other algebraic structures, advancing the understanding of Lie-Yamaguti algebra deformations.

Contribution

It establishes the cohomology framework for LieYRep pairs and introduces the concept of relative Rota-Baxter-Nijenhuis structures, linking them to compatible Rota-Baxter operators and $r$-matrix-Nijenhuis structures.

Findings

Cohomology theory characterizes linear deformations of LieYRep pairs.

Relative Rota-Baxter-Nijenhuis structures induce compatible Rota-Baxter operators.

Equivalence between $r$-matrix-Nijenhuis and Rota-Baxter-Nijenhuis structures on Lie-Yamaguti algebras.

Abstract

A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In this paper, we establish the cohomology theory of LieYRep pairs and characterize their linear deformations by the second cohomology group. Then we introduce the notion of relative Rota-Baxter-Nijenhuis structures on LieYRep pairs, investigate their properties, and prove that a relative Rota-Baxter-Nijenhuis structure gives rise to a pair of compatible relative Rota-Baxter operators under a certain condition. Finally, we show the equivalence between $r$-matrix-Nijenhuis structures and Rota-Baxter-Nijenhuis structures on Lie-Yamaguti algebras.