Finiteness and dimension of stated skein modules over Frobenius

TL;DR

This paper proves that stated skein modules at roots of unity are finitely generated and finite dimensional over certain base modules, providing bounds on their dimensions and exploring their algebraic structure.

Contribution

It establishes finiteness and dimension bounds for stated skein modules at roots of unity, and analyzes their algebraic properties related to Frobenius maps.

Findings

Skein modules are finitely generated over base modules when the quantum parameter is a root of unity.

The reduced skein module for compact manifolds is finite dimensional.

The dimension of certain field extensions of skein algebras is explicitly calculated as N^{3r(Σ)}.

Abstract

When the quantum parameter $q^{1/2}$ is a root of unity of odd order. The stated skein module $S_{q^{1/2}}(M,\mathcal{N})$ has an $S_{1}(M,\mathcal{N})$-module structure, where $(M,\mathcal{N})$ is a marked three manifold. We prove $S_{q^{1/2}}(M,\mathcal{N})$ is a finitely generated $S_{1}(M,\mathcal{N})$-module when $M$ is compact, which furthermore indicates the reduced stated skein module for the compact marked three manifold is finite dimensional. We also give an upper bound for the dimension of $S_{q^{1/2}}(M,\mathcal{N})$ over $S_{1}(M,\mathcal{N})$ when $M$ is compact. For a pb surface $\Sigma$, we use $S_{q^{1/2}}(\Sigma)^{(N)}$ to denote the image of the Frobenius map when $q^{1/2}$ is a root of unity of odd order $N$. Then $S_{q^{1/2}}(\Sigma)^{(N)}$ lives in the center of the stated skein algebra $S_{q^{1/2}}(\Sigma)$. Let $\widetilde{S_{q^{1/2}}(\Sigma)^{(N)}}$ be the field of fractions of $S_{q^{1/2}}(\Sigma)^{(N)}$, and $\widetilde{S_{q^{1/2}}(\Sigma)}$ be $S_{q^{1/2}}(\Sigma)\otimes_{S_{q^{1/2}}(\Sigma)^{(N)}} \widetilde{S_{q^{1/2}}(\Sigma)^{(N)}}$. Then we show the dimension of $\widetilde{S_{q^{1/2}}(\Sigma)}$ over $\widetilde{S_{q^{1/2}}(\Sigma)^{(N)}}$ is $N^{3r(\Sigma)}$ where $r(\Sigma)$ equals to the number of boundary components of $\Sigma$ minus the Euler characteristic of $\Sigma$.