Stated SL(n)-skein modules and algebras

We develop a theory of stated SL((Formula presented.))-skein modules, (Formula presented.), of 3-manifolds (Formula presented.) marked with intervals (Formula presented.) 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 (Formula presented.) -webs, with ends in (Formula presented.), considered up to skein relations of the (Formula presented.) -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 (Formula presented.) along a disk resulting in a 3-manifold (Formula presented.) yields a homomorphism (Formula presented.) for all (Formula presented.). 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 (Formula presented.), in whose case, (Formula presented.) is an algebra, denoted by (Formula presented.). One of the main results of this paper asserts that the skein algebra of the ideal bigon is isomorphic with (Formula presented.) and it provides simple geometric interpretations of the product, coproduct, counit, the antipode, and the cobraided structure on (Formula presented.). (In particular, the coproduct is given by a splitting homomorphism.) We show that for surfaces with boundary (Formula presented.) every splitting homomorphism is injective and that (Formula presented.) 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 (Formula presented.) -comodule structure on (Formula presented.), or dually, a left (Formula presented.) -module structure. Furthermore, we show that the skein algebra of surfaces (Formula presented.) glued along two sides of a triangle is isomorphic with the braided tensor product (Formula presented.) 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 (Formula presented.) is equivalent to the category of left modules over (Formula presented.) for surfaces (Formula presented.) with (Formula presented.). 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 (Formula presented.).