Deformations and Cohomology Theory of Rota-Baxter Systems

TL;DR

This paper develops a cohomology theory for Rota-Baxter systems, linking it to deformations and extensions, and sets the stage for future work on related algebraic structures.

Contribution

It introduces a cohomology framework for Rota-Baxter systems, connecting cohomology groups to deformations and extensions, expanding the algebraic understanding of these systems.

Findings

Lower degree cohomology groups correspond to formal deformations.

Cohomology groups classify abelian extensions of Rota-Baxter systems.

Foundation laid for future study of $L_\infty$-algebra structures.

Abstract

Inspired by the work of Wang and Zhou [4] for Rota-Baxter algebras, we develop a cohomology theory of Rota-Baxter systems and justify it by interpreting the lower degree cohomology groups as formal deformations and as abelian extensions of Rota-Baxter systems. A further study on an $L_\infty$-algebra structure associated to this cohomology theory will be given in a subsequent paper.