Center of generalized skein algebras

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.

💡 Research Summary

The paper studies the algebraic structure of the Muller‑Roger‑Yang (MRY) skein algebra (S^{\mathrm{MRY}}_{q}(\Sigma)), a generalization of the Kauffman bracket skein algebra that incorporates both loops and arcs whose endpoints may lie at interior punctures or marked points on the boundary of a surface (\Sigma). The authors’ main achievement is a complete description of the center of this algebra when the quantum parameter (q) is a primitive (n)-th root of unity with odd (n), together with a proof that the algebra is “almost Azumaya” under the same hypothesis.

Background and definitions.
A marked surface ((\Sigma,V)) consists of a compact oriented surface (\Sigma) together with a finite set (V=V^{\circ}\sqcup V^{\partial}) of interior punctures and boundary marked points. The MRY skein algebra is generated by isotopy classes of (V)-tangles (embedded framed 1‑manifolds) modulo the usual Kauffman bracket relations (A)–(C), together with puncture‑skein relations (D)–(F) that involve the formal variables (v_i) attached to each interior puncture. The coefficient ring is (\mathbb{C}