The classical limit for stated $SL_n$-skein modules

Abstract

Let $(M,\mathcal{N})$ be a marked 3-manifold. We use $S_n(M,\mathcal{N},v)$ to denote the stated $SL_n$-skein module of $(M,\mathcal{N})$ where $v$ is a nonzero complex number. We establish a surjective algebra homomorphism from $S_n(M,\mathcal{N},1)$ to the coordinate ring of some algebraic set, and prove its kernel consists of all nilpotents. We prove the universal representation algebra of $π_1(M)$ is isomorphic to $S_n(M,\mathcal{N},1)$ when $M$ is connected and $\mathcal{N}$ has only one component. Furthermore, we show $S_n(M,\mathcal{N}',1)$ is isomorphic to $S_n(M,\mathcal{N},1)\otimes O(SL_n)$ as algebras, where $(M,\mathcal{N})$ is a connected marked 3-manifold with $\mathcal{N}\neq\emptyset$, and $\mathcal{N}'$ is obtained from $\mathcal{N}$ by adding one extra marking.

Figures (1)

• Figure 1: The orientation of $D$ is indicated by the arrow, the orientation of $M$ is right handed. The left (respectively right) disk copy is $D_1$ (respectively $D_2$). The left (respectively right) red arrow is $\beta_1$ (respectively $\beta_2$).

Theorems & Definitions (85)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1: le2021stated
• Corollary 2.2: le2021stated
• Lemma 3.1
• proof
• Proposition 3.2