Cohomologies and linear deformations of relative Rota-Baxter operators on (pre-)Jacobi-Jordan algebras

TL;DR

This paper develops a cohomology theory for relative Rota-Baxter operators on (pre-)Jacobi-Jordan algebras, introducing concepts like matched pairs, Nijenhuis elements, and analyzing linear deformations.

Contribution

It introduces the cohomology framework for these operators and explores their deformations, providing new tools for understanding their structure and triviality conditions.

Findings

Cohomology theory for relative Rota-Baxter operators is established.

Nijenhuis elements are characterized to identify trivial deformations.

Linear deformations are systematically studied using the cohomological approach.

Abstract

Some results on (pre-)Jacobi-Jordan algebras and their representations are proved. Moreover, the notion of matched pairs and relative Rota-Baxter operators on these algebras are introduced and studied. The cohomology theory of relative Rota-Baxter operators on (pre)-Jacobi-Jordan algebras is introduced. We use the cohomological approach to study linear deformations of relative Rota-Baxter operators. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations.