The quantum Frobenius for character varieties and multiplicative quiver varieties

TL;DR

This paper establishes a deep connection between quantum and classical moduli spaces through Azumaya algebras, proving the Azumaya locus contains the smooth locus and advancing the understanding of quantum character and multiplicative quiver varieties.

Contribution

It introduces Frobenius quantum moment maps, Frobenius Poisson orders, and demonstrates their role in relating quantum and classical structures, proving the Azumaya property in this context.

Findings

Quantized multiplicative quiver varieties form sheaves of Azumaya algebras.

The Azumaya locus of Kauffman bracket skein algebras contains the smooth locus.

New tools like Frobenius quantum moment maps are developed and applied.

Abstract

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.