Center of generalized skein algebras

TL;DR

This work determines the center of the Muller-Roger-Yang generalized skein algebra $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ for marked surfaces with interior punctures and boundary markings. It proves that at odd root-of-unity order $n$, the center is generated by Chebyshev-type threadings of loops and certain arc classes, together with boundary components, and establishes a square-trick and generalized edge coordinates to structurally control central elements. A central technical contribution is the center characterization via edge coordinates and leading-term comparisons, building on the Chebyshev–Frobenius framework and results from related skein algebras. The paper further shows that $\mathcal{S}_{q}^{\mathrm{MRY}}(\Sigma)$ is almost Azumaya, enabling a controlled description of finite-dimensional irreducible representations through the Azumaya locus and connecting these algebraic findings to quantum cluster algebra perspectives and representation theory of skein-type algebras.

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.

Figures (10)

• Figure 1: Reidemeister moves
• Figure 2: An example of moving trick. The left $V$-tangle diagram does not have the preferred crossing because the right tangle diagram has an extra crossing. The sequence under the $V$-tangle diagram denotes the height order of three strands ($1$ is the highest end.).
• Figure 3: A bad arc
• Figure 4: Left: (partial) picture of $\gamma$, Middle: (partial) picture of $R$, Right: (partial) picture of $L$, Outside the shaded part, the three arcs are the same.
• Figure 5: Comparison of $f(\alpha_{1}) + f(\alpha_{2})$ and resolutions of $v\alpha_{1}\alpha_{2}$. The second row shows two resolutions of $v\alpha_{1}\alpha_{2}$ and their edge coordinates.

Theorems & Definitions (52)

• Theorem A
• Corollary 1.1
• Definition 2.1: The Muller--Roger--Yang skein algebra BKL24
• Remark 2.2
• Definition 2.3: The Lê--Roger--Yang skein algebra BKL24
• Remark 2.4
• Theorem 2.5: BKL24
• Remark 2.6
• Definition 2.7
• Proposition 2.8: BKL24