Cohomology and Deformations of Twisted Rota-Baxter Family Operators on Hom-Associative Algebras

In this paper, we first define twisted Rota-Baxter family operators on Hom-associative algebras indexed by a semigroup $$\mho $$ . We then introduce Hom-NS-family algebras as structures for twisted Rota-Baxter family operators. Meanwhile, we demonstrate that a Hom-NS-family algebra naturally induces a Hom-NS-algebra. Moreover, we establish the cohomology of twisted Rota-Baxter family operators. This cohomology may also be interpreted as the cohomology of a certain Hom- $$\mho $$ -associative algebra with coefficients in a 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.