Let $k$ be a field of characteristic zero, $\mathfrak{g}$ a finite-dimensional Lie algebra over $k$, and let $A,B$ associative $k$-algebras.

Suppose that $\mathfrak{g}$ acts on $A$ and $B$, and that each of these actions admit a quantum moment map. Then the associative algebra $A\otimes_kB$ also has a $\mathfrak{g}$ action and admits a quantum moment map.

In this case, is it true that

$$(A\otimes_kB)//\mathfrak{g}\cong (A//\mathfrak{g})\otimes_k(B//\mathfrak{g})$$

where "$//\mathfrak{g}$" is the operation of quantum Hamiltonian reduction.

Let $k$ be a field of characteristic zero, $\mathfrak{g}$ a finite-dimensional Lie algebra over $k$, and let $A,B$ associative $k$-algebras.

Suppose that $\mathfrak{g}$ acts on $A$ and $B$, and that each of these actions admit a quantum moment map. Then the associative algebra $A\otimes_kB$ also has a $\mathfrak{g}$ action and admits a quantum moment map.

In this case, is it true that

$$(A\otimes_kB)//\mathfrak{g}\cong (A//\mathfrak{g})\otimes_k(B//\mathfrak{g})$$

where "$//\mathfrak{g}$" is the operation of quantum Hamiltonian reduction?