The $q$-analog of higher order Hochschild homology and the Lie derivative

TL;DR

This paper introduces a $q$-analog of higher order Hochschild homology for commutative algebras, incorporating Lie derivatives and higher derivations, and explores their effects on bivariant cohomology groups.

Contribution

It defines $q$-Hochschild homology for higher orders and constructs Lie derivatives for derivations and higher derivations, extending classical concepts.

Findings

Defined $q$-Hochschild homology groups of order $Y$ for commutative algebras.

Constructed Lie derivatives on these homology groups for derivations and higher derivations.

Described morphisms on bivariant $q$-Hochschild cohomology induced by derivations.

Abstract

Let $A$ be a commutative algebra over $\mathbb C$. Given a pointed simplicial finite set $Y$ and $q\in \mathbb C$ a primitive $N$-th root of unity, we define the $q$-Hochschild homology groups of $A$ of order $Y$. When $D$ is a derivation on $A$, we construct the corresponding Lie derivative on these groups. We also define the Lie derivative for a higher derivation $\{D_n\}_{n\geq 0}$ on $A$. Finally, we describe the morphisms induced on the bivariant $q$-Hochschild cohomology groups of order $Y$ by a derivation $D$ on $A$.