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

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 $A$.