Cohomology theory of Rota-Baxter family BiHom-$\Omega$-associative algebras

TL;DR

This paper develops a cohomology theory for Rota-Baxter family BiHom-$\Omega$-associative algebras, establishing its role in controlling deformations and classifying abelian extensions.

Contribution

It introduces the cohomology framework for these complex algebras and demonstrates its applications in deformation theory and extension classification.

Findings

Defined the cochain complex for BiHom-$\Omega$-associative algebras.

Established the cohomology controls algebra deformations.

Connected second cohomology to abelian extensions.

Abstract

In this paper, we first introduce the concept of Rota-Baxter family BiHom-$\Omega$-associative algebras, then we define the cochain complex of BiHom-$\Omega$-associative algebras and verify it via Maurer-Cartan methods. Next, we further introduce and study the cohomology theory of Rota-Baxter family BiHom-$\Omega$-associative algebras of weight $\lambda$ and show that this cohomology controls the corresponding deformations. Finally, we study abelian extensions of Rota-Baxter family BiHom-$\Omega$-associative algebras in terms of the second cohomology group.