Stated SL(n)-Skein Modules and Algebras

We develop a theory of stated SL (n) -skein modules, Sₙ (M, N), of 3-manifolds M marked with intervals N in their boundaries. They consist of linear combinations of n-webs with ends in N, considered up to skein relations inspired by the relations of the Reshetikhin-Turaev theory. We prove that cutting M along a disk resulting in a 3-manifold M' yields a homomorphism Sₙ (M) Sₙ (M'). 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=Σ (-1, 1), in whose case, Sₙ (M) is an algebra, denoted by Sₙ (Σ). We prove that the skein algebra of the ideal bigon is Oq (SL (n) ) and that it provides simple geometric interpretations of the product, coproduct, counit, the antipode, and the cobraided structure on Oq (SL (n) ). Additionally, we show that a splitting of a thickened bigon near a marking defines a Oq (SL (n) ) -comodule structure on Sₙ (M), or dually, an Uq (slₙ) -module structure. Furthermore, we show that the skein algebra of surfaces Σ₁, Σ₂ glued along two sides of a triangle is isomorphic with the braided tensor product Sₙ (Σ₁) Sₙ (Σ₂) of Majid. These results allow for a geometric interpretation of further concepts in the theory of quantum groups, for example, of the braided products and of Majid's transmutation operation. We prove that the factorization homology of surfaces with coefficients in Rep\, Uq (slₙ) is equivalent to the category of left modules over Sₙ (Σ). We also discuss the relation with the quantum moduli spaces of Alekseev-Schomerus. Finally, we show that for surfaces Σ with boundary, Sₙ (Σ) is a free module with a basis induced from the Kashiwara-Lusztig canonical bases.