On stated $SL(n)$-skein modules

Abstract

We mainly focus on Classical limit, Splitting map, and Frobenius homomorphism for stated $SL(n)$-skein modules, and Unicity Theorem for stated $SL(n)$-skein algebras. Let $(M,N)$ be a marked three manifold. We use $S_n(M,N,v)$ to denote the stated $SL(n)$-skein module of $(M,N)$ where $v$ is a nonzero complex number. We build a surjective algebra homomorphism from $S_n(M,N,1)$ to the coordinate ring of some algebraic set, and prove it's Kernal consists of all nilpotents. We prove the universal representation algebra of $\pi_1(M)$ is isomorphic to $S_n(M,N,1)$ when $N$ has only one component and $M$ is connected. Furthermore we show $S_n(M,N^{'},1)$ is isomorphic to $S_n(M,N,1)\otimes O(SLn)$, where $N\neq \emptyset$, $M$ is connected, and $N^{'}$ is obtained from $N$ by adding one extra marking. We also prove the splitting map is injective for any marked three manifold when $v=1$, and show that the splitting map is injective (for general $v$) if there exists at least one component of $N$ such that this component and the boundary of the splitting disk belong to the same component of $\partial M$. We also establish the Frobenius homomorphism for $SL(n)$, which is map from $S_n(M,N,1)$ to $S_n(M,N,v)$ when $v$ is a primitive $m$-th root of unity with $m$ being coprime with $2n$ and every component of $M$ contains at least one marking. We also show the commutativity between Frobenius homomorphism and splitting map. When $(M,N)$ is the thickening of an essentially bordered pb surface, we prove the Frobenius homomorphism is injective and it's image lives in the center. We prove the stated $SL(n)$-skein algebra $S_n(\Sigma,v)$ is affine almost Azumaya when $\Sigma$ is an essentially bordered pb surface and $v$ is a primitive $m$-th root of unity with $m$ being coprime with $2n$, which implies the Unicity Theorem for $S_n(\Sigma,v)$.