Cohomology theory of Nijenhuis family $\Omega$-associative algebras

TL;DR

This paper introduces Nijenhuis family $ abla$-associative algebras, develops their cohomology theory, and explores their deformations and extensions, linking algebraic structures with quantum field theory applications.

Contribution

It defines Nijenhuis family $ abla$-associative algebras, establishes their cohomology, and connects these concepts to algebraic deformations and extensions.

Findings

Cohomology controls algebra deformations.

Second cohomology classifies abelian extensions.

Relationship established between Nijenhuis operators and $ abla$-associative structures.

Abstract

Family algebraic structures indexed by a semigroup first appeared in the algebraic aspects of renormalizations in quantum field theory. In this paper, we first introduce the concept of Nijenhuis family $\Omega$-associative algebras and we discuss the relationship between Nijenhuis family and other operators families on $\Omega$-associative algebras. Then, we define the cohomology theory of Nijenhuis family $\Omega$-associative algebras and show that this cohomology controls the corresponding deformations. Finally, we study abelian extensions of Nijenhuis family $\Omega$-associative algebras in terms of the second cohomology group.