Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
18
The categories of this paper are: econ
23
The categories of this paper are: solv-int
21
The categories of this paper are: adap-org
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph