Center of generalized skein algebras

TL;DR

This paper generalizes the skein algebra of a surface to include arcs and loops, computes its center at roots of unity, and explores implications for its representation theory.

Contribution

It computes the center of the Muller-Roger-Yang skein algebra and analyzes its structure at roots of unity, extending previous skein algebra results.

Findings

Center is almost Azumaya at roots of unity.

Provides explicit description of the algebra's center.

Discusses implications for representation theory.

Abstract

We consider a generalization of the Kauffman bracket skein algebra of a surface that is generated by loops and arcs between marked points on the interior or boundary, up to skein relations defined by Muller and Roger-Yang. We compute the center of this Muller-Roger-Yang skein algebra and show that it is almost Azumaya when the quantum parameter $q$ is a primitive $n$-th root of unity with odd $n$. We also discuss the implications on the representation theory of the Muller-Roger-Yang generalized skein algebra.