Additive and multiplicative quiver varieties are certain representation theoretic moduli spaces with rich geometry: they are holomorphic symplectic, and give basic examples of symplectic resolutions. They are constructed by a procedure called Hamiltonian reduction, which allows one to study their geometry via the representation theory of the gauge group. Examples include flag varieties in type $A$, Calogero-Moser (additive) and Ruijsenaars-Schneider (multiplicative) integrable systems, Deligne-Simpson moduli spaces of flat connections (additive) and character varieties (multiplicative) of punctured surfaces.
Several years ago I introduced quantizations of multiplicative quiver varieties, which deform the symplectic form into a non-commutative algebra. The gauge group is replaced by the corresponding quantum group $U_q(\mathfrak{g})$, so that the quantizations naturally depend on a complex parameter $q$. I'll recall this construction, and report on joint work with Iordan Ganev which considers the very special situation when $q$ is a root of unity. There we observe what we call "quantum Frobenius" phenomenon: all the non-commutative algebras in the room develop large centers, which in fact identify with a "Frobenius twist" of the classical multiplicative quiver variety. Moreover, the whole non-commutative algebra gives rise to a generically Azumaya algebra: meaning that \'etale-locally (but not globally) over some open locus on the base, it is isomorphic to a simple matrix algebra. Finally, I'll explain a conjecture pinning down the locus where this holds, following ideas of Brown and Yakimov.