Twisted Rota-Baxter family operators on Hom-associative algebras

TL;DR

This paper introduces twisted Rota-Baxter family operators on Hom-associative algebras, explores their underlying Hom-NS-family structures, develops their cohomology, and studies their deformations and rigidity conditions.

Contribution

It defines twisted Rota-Baxter family operators on Hom-associative algebras and investigates their cohomology, deformations, and rigidity, which are novel contributions in this area.

Findings

Hom-NS-family algebras induce ordinary Hom-NS-algebras on tensor products.

Cohomology of twisted Rota-Baxter operators is established and linked to Hom-$\

Deformations are governed by the developed cohomology, and Nijenhuis elements provide rigidity conditions.

Abstract

In this paper, we first define twisted Rota-Baxter family operators on Hom-associative algebras indexed by a semigroup $\Omega$. Then we introduce and study Hom-NS-family algebras as the underlying structures of twisted Rota-Baxter family operators. Meanwhile, We show that a Hom-NS-family algebra induces an ordinary Hom-NS-algebra on the tensor product with the semigroup algebra. Moreover, we define the cohomology of a twisted Rota-Baxter family operator. This cohomology can also be viewed as the cohomology of a certain Hom-$\Omega$-associative algebra with coefficients in a suitable bimodule. Finally, we examine deformations of twisted Rota-Baxter family operators and demonstrate that they are governed by the aforementioned cohomology. The concept of Nijenhuis elements linked to a twisted Rota-Baxter family operator is introduced to provide a sufficient condition for its rigidity.