Universal $q$-differential calculus and $q$-analog of homological algebra

TL;DR

This paper develops a universal framework for $q$-differential calculus, introducing a $q$-analog of homological algebra concepts like Hochschild cohomology, and explores properties of $d^N=0$ in this context.

Contribution

It constructs the universal $q$-differential envelope of unital associative algebras and extends homological algebra to the $q$-deformed setting.

Findings

Construction of the $q$-analog of Hochschild coboundary

Development of the universal $q$-differential envelope

Results on properties of $d^N=0$ in $q$-differential calculus

Abstract

We recall the definition of $q$-differential algebras and discuss some representative examples. In particular we construct the $q$-analog of the Hochschild coboundary. We then construct the universal $q$-differential envelope of a unital associative algebra and study its properties. The paper also contains general results on $d^N=0$.