Representation-reduced stated skein modules and algebras

TL;DR

This paper studies the structure and representations of stated skein modules of 3-manifolds at roots of unity, proving splitting map properties, irreducibility, and dimension results for reduced modules.

Contribution

It introduces a detailed analysis of representation-reduced skein modules, establishing their irreducibility and dimension, and clarifies the splitting map behavior at roots of unity.

Findings

Splitting map respects module structure and is injective under certain boundary conditions.

Representation-reduced skein modules of handlebodies are irreducible Azumaya algebras.

Dimensions of reduced skein modules are equal to one for certain marked manifolds.

Abstract

For any marked three manifold $(M,\mathcal N)$ and any quantum parameter $q^{\frac{1}{2}}$ (a nonzero complex number), we use $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})$ to denote the stated skein module of $(M,\mathcal{N})$. When $q^{\frac{1}{2}}$ is a root of unity of odd order, the commutative algebra $\mathscr{S}_1(M,\mathcal{N})$ acts on $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})$. For any maximal ideal $\rho$ of $\mathscr{S}_1(M,\mathcal{N})$, define $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})_{\rho} = \mathscr{S}_{q^{1/2}}(M,\mathcal{N})\otimes _{\mathscr{S}_1(M,\mathcal{N})} (\mathscr{S}_1(M,\mathcal{N})/\rho)$. We prove the splitting map for $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})$ respects the $\mathscr{S}_1(M,\mathcal{N})$-module structure, so it reduces to the splitting map for $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})_{\rho}$. We prove the splitting map for $\mathscr{S}_{q^{1/2}}(M,\mathcal{N})_{\rho}$ is injective if there exists at least one component of $\mathcal{N}$ such that this component and the boundary of the splitting disk belong to the same component of $\partial M$. We also prove the representation-reduced stated skein module of the marked handlebody is an irreducible Azumaya representation of the stated skein algebra of its boundary. Let $M$ be an oriented connected closed three manifold. For any positive integer $k$, we use $M_{k}$ to denote the marked three manifold obtained from $M$ by removing $k$ open three dimensional balls and adding one marking to each newly created sphere boundary component. We prove $\text{dim}_{\mathbb{C}}\mathscr{S}_{q^{1/2}}( M_{k})_{\rho} = 1$ for any maximal ideal $\rho$ of $\mathscr{S}_1(M_k)$.