  We prove that quantized multiplicative quiver varieties and quantum character
varieties define sheaves of Azumaya algebras over the corresponding classical
moduli spaces, and we prove that the Azumaya locus of the Kauffman bracket
skein algebras contains the smooth locus, proving a strong form of the Unicity
Conjecture of Bonahon and Wong. The proofs exploit a strong compatibility
between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as
it arises in each setting. We therefore introduce the concepts of Frobenius
quantum moment maps and their Hamiltonian reduction, and of Frobenius Poisson
orders. We use these tools to construct canonical central subalgebras of
quantum algebras, and explicitly compute the resulting Azumaya loci we
encounter, using a natural nondegeneracy assumption.
