Stated SL( n $n$ )-skein modules and algebras

Abstract

We develop a theory of stated SL($n$)-skein modules, $S_n(M,N)$, of 3-manifolds $M$ marked with intervals $N$ in their boundaries. These skein modules, generalizing stated SL(2)-modules of the first author, stated SL(3)-modules of Higgins', and SU(n)-skein modules of the second author, consist of linear combinations of framed, oriented graphs, called $n$-webs, with ends in $N$, considered up to skein relations of the $U_q\left(s{l}_n\right)$-Reshetikhin–Turaev functor on tangles, involving coupons representing the anti-symmetrizer and its dual. We prove the Splitting Theorem asserting that cutting of a marked 3-manifold $M$ along a disk resulting in a 3-manifold $M^{\prime }$ yields a homomorphism $S_n\left(M\right)\to S_n\left(M^{\prime }\right)$ for all $n$. That result allows to analyze the skein modules of 3-manifolds through the skein modules of their pieces. The theory of stated skein modules is particularly rich for thickened surfaces $M=\Sigma \times (-1,1)$, in whose case, $S_n\left(M\right)$ is an algebra, denoted by $S_n\left(\Sigma \right)$. One of the main results of this paper asserts that the skein algebra of the ideal bigon is isomorphic with $O_q\left(SL\left(n\right)\right)$ and it provides simple geometric interpretations of the product, coproduct, counit, the antipode, and the cobraided structure on $O_q\left(SL\left(n\right)\right)$. (In particular, the coproduct is given by a splitting homomorphism.) We show that for surfaces with boundary $\Sigma$ every splitting homomorphism is injective and that $S_n\left(\Sigma \right)$ is a free module with a basis induced from the Kashiwara–Lusztig canonical bases. Additionally, we show that a splitting of a thickened bigon near a marking defines a right $O_q\left(SL\left(n\right)\right)$-comodule structure on $S_n\left(M\right)$, or dually, a left $U_q\left(s{l}_n\right)$-module structure. Furthermore, we show that the skein algebra of surfaces ${\Sigma }_1,{\Sigma }_2$ glued along two sides of a triangle is isomorphic with the braided tensor product $S_n\left({\Sigma }_1\right)\underline{\otimes }{S}_n\left({\Sigma }_2\right)$ of Majid. These results allow for geometric interpretation of further concepts in the theory of quantum groups, for example, of the braided products and of Majid's transmutation operation. Building upon the above results, we prove that the factorization homology with coefficients in the category of representations of $U_q\left(s{l}_n\right)$ is equivalent to the category of left modules over $S_n\left(\Sigma \right)$ for surfaces $\Sigma$ with $\partial \Sigma =S^1$. We also establish isomorphisms of our skein algebras with the quantum moduli spaces of Alekseev–Schomerus and with the internal algebras of the skein categories for these surfaces and $g=sl\left(n\right)$.