Let $k$ be a field of characteristic 0 and $A$ a unital associative $k$-algebra.

The representation functor associates, to each object in non-commutative geometry, a genuine geometric object on the representation variety $\mathrm{Rep}(A, V) = \mathrm{Hom}_\mathrm{alg}(A,\mathrm{End}(V))$ of $A$ on a finite-dimensional $k$-vector space $V$. For example, we have a map on the space of functions: $A \to \mathcal{O}(\mathrm{Rep}(A,V))\otimes_k \mathrm{End}(V)$ given by $a\mapsto (\rho \mapsto \rho(a))$. Many more are listed in Section 12 of Ginzburg's lecture note.

Question: How can we construct the corresponding commutative object to: (1) an $A$-(bi)module, and (2) a connection on $M$ (e.g., of the form $\nabla\colon M \to \Omega^1A \otimes_A M$ in one formulation)?

The space of double derivations is worked out there, but their construction does not apply to arbitrary modules. Any suggestions or comments are appreciated.