The q-connections

Question

Let $(M,q)$ be a manifold with an action of $q$. A $q$-derivation $X$ is such that:

$$X(fg)(x)=X(f)(x)g(qx)+f(x)X(g)(x)$$

A $q$-connection $\nabla$ is such that:

$$\nabla_X (fs)(x)=X(f)(x) s(qx)+f(x)\nabla_X (s)(x)$$

where $X$ is a $q$-derivation. Can we make quantum differential geometry with these notions?