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Twisted Rota-Baxter families and NS-family algebras

Family algebraic structures indexed by a semigroup first appeared in the algebraic aspects of renormalizations in quantum field theory. The concept of the Rota-Baxter family and its relation with (tri)dendriform family algebras have been recently discovered. In this paper, we first consider a notion of $\mathcal{O}$-operator family as a generalization of the Rota-Baxter family and define two variations of associative Yang-Baxter family that produce $\mathcal{O}$-operator families. Given a Hochschild $2$-cocycle on the underlying algebra, we also define a notion of twisted $\mathcal{O}$-operator family (in particular twisted Rota-Baxter family). We also introduce and study NS-family algebras as the underlying structure of twisted $\mathcal{O}$-operator families. Finally, we define suitable cohomology of twisted $\mathcal{O}$-operator families and NS-family algebras (in particular cohomology of Rota-Baxter families and dendriform family algebras) that govern their deformations.