2019/07/26 by Costantino, Francesco, Le, Thang T. Q. · 2 citations
#FOS: Mathematics #Geometric Topology (math.GT) #Primary 57N10 #secondary 57M25
paper · doi:10.48550/arxiv.1907.11400
We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group \mathcal Oq2(SL(2)) providing a topological interpretation for its structure morphisms. We also show that its stated skein algebra lifts in a suitable sense the Reshetikhin-Turaev functor and in particular we recover the dual R-matrix for \mathcal Oq2(SL(2)) in a topological way. We deduce that the skein algebra of a surface with n boundary components is an algebra-comodule over \mathcal Oq2(SL(2))^⊗n and prove that cutting along an ideal arc corresponds to Hochshild cohomology of bicomodules. We give a topological interpretation of braided tensor product of stated skein algebras of surfaces as "glueing on a triangle"; then we recover topologically some braided bialgebras in the category of \mathcal Oq2(SL(2))-comodules, among which the "transmutation" of \mathcal Oq2(SL(2)). We also provide an operadic interpretation of stated skein algebras as an example of a "geometric non symmetric modular operad". In the last part of the paper we define a reduced version of stated skein algebras and prove that it allows to recover Bonahon-Wong's quantum trace map and interpret skein algebras in the classical limit when q→ 1 as regular functions over a suitable version of moduli spaces of twisted bundles.