Stated skein modules of marked 3-manifolds/surfaces, a survey

TL;DR

This survey reviews the structure, homomorphisms, and representation theory of stated skein modules of 3-manifolds and surfaces, highlighting results at generic quantum parameters and roots of unity.

Contribution

It provides a comprehensive overview of the algebraic structures, homomorphisms, and representation theory of stated skein modules, including new results on their centers and Azumaya loci.

Findings

Splitting homomorphism for 3-manifolds

Embedding of skein algebras into quantum tori

Description of the center and Azumaya locus at roots of unity

Abstract

We give a survey of some old and new results about the stated skein modules/algebras of 3-manifolds/surfaces. For generic quantum parameter, we discuss the splitting homomorphism for the 3-manifold case, general structures of the stated skein algebras of marked surfaces (or bordered punctured surfaces) and their embeddings into quantum tori. For roots of 1 quantum parameter, we discuss the Frobenius homomorphism (for both marked 3-manifolds and marked surfaces), describe the center of the skein algebra of marked surfaces, the dimension of the skein algebra over the center, and the representation theory of the skein algebra. In particular, we show that the skein algebra of non-closed marked surface at any root of 1 is a maximal order. We give a full description of the Azumaya locus of the skein algebra of the puncture torus and give partial results for closed surfaces.