STATED SKEIN ALGEBRAS AND A GEOMETRIC APPROACH TO QUANTUM GROUPS

We introduce the theory of stated SL(n)-skein algebras of surfaces, which provide a geometric/combinatorial interpretation for the quantum groups Oq(sl(n)) and other related notions from quantum algebra. They also quantize the SL(n)-character varieties of surfaces, are examples of quantum cluster algebras, and are closely related to Reshetikhin-Turaev quantum invariants of links, factorization homology, and lattice gauge theory.