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 $\pi_1(M)$ 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.