Let $A$ be an associative algebra over a field $k$. Let $A_{L}$ be the Lie algebra of $A$ with commutator bracket. Then if $M$ is a bimodule for $A$ there is an associated representation of $A_{L}$ called the adjoint representation and denoted $M^{adj}$.

There is a naturally defined map of homology groups $H^{Lie} _{*} (A_{Lie}, M^{adj}) \rightarrow HH_{*} (A, M).$ This is defined by anti-symmetrizing a Lie chain.

I'd like to know when this map on homology is surjective. It is if $A$ is smooth commutative (this follows from HKR). Less obviously it is true for $U\mathfrak{g}$, as follows from a computation of Chevalley and Eilenberg. A sort of minimal possible generalization would perhaps be to an almost commutative algebra, ie an associative algebra with an increasing filtration whose associated graded is smooth commutative. I'd love if it were true in this generality but any partial results would be greatly appreciated.