  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
$A$.
