Noetherian and affine properties of quantum moduli and $\mathfrak{g}$-skein algebras

TL;DR

This paper establishes that quantum moduli algebras for punctured surfaces are Noetherian, finitely generated, and domains, and shows their isomorphism with skein algebras derived from quantum groups, extending known results for special cases.

Contribution

It proves the Noetherian and domain properties of quantum moduli algebras for punctured surfaces and establishes their isomorphism with skein algebras, generalizing previous results.

Findings

Quantum moduli algebra is Noetherian and finitely generated.

Quantum moduli algebra is a domain if the surface has punctures.

Quantum moduli algebra is isomorphic to the skein algebra of the surface.

Abstract

We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra $\mathfrak{g}$ is a Noetherian and finitely generated ring. If the surface has punctures, we prove also that it has no non-trivial zero divisors (i.e., it is a domain). Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin-Turaev functor for the quantum group $U_q(\mathfrak{g})$, and which coincides with the Kauffman bracket skein algebra when $\mathfrak{g}=\mathfrak{sl}_2$. We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras.