Noncommutative differential calculus on (co)homology of hom-associative algebras

TL;DR

This paper demonstrates that the Hochschild (co)homology of hom-associative algebras admits a noncommutative differential calculus structure, extending known algebraic structures and leading to a Batalin-Vilkovisky algebra in specific cases.

Contribution

It introduces a noncommutative differential calculus framework on Hochschild (co)homology of hom-associative algebras, including new algebraic structures and applications.

Findings

Hochschild cohomology of hom-associative algebras has a Gerstenhaber structure.

The Hochschild homology admits operations forming a noncommutative differential calculus.

A Batalin-Vilkovisky algebra structure is obtained for certain hom-associative algebras.

Abstract

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. It was previously shown by the author that the Hochschild cohomology of a hom-associative algebra $A$ carries a Gerstenhaber structure. In this short paper, we show that this Gerstenhaber structure together with certain operations on the Hochschild homology of $A$ makes a noncommutative differential calculus. As an application, we obtain a Batalin-Vilkovisky algebra structure on the Hochschild cohomology of a regular unital symmetric hom-associative algebra.