Stated skein algebras of surfaces

Abstract

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 $\mathcal{O}_{{q^2}}(\operatorname{SL}(2))$ thus 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 $\mathcal{O}_{{q^2}}(\operatorname{SL}(2)$ in a topological way. We deduce that the skein algebra of a surface with $n$ boundary components is a comodule algebra over $\mathcal{O}_{{q^2}}(\operatorname{SL}(2))^{\otimes n}$ 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 “gluing on a triangle”; then we recover topologically some bialgebras in the category of $\mathcal{O}_{{q^2}}(\operatorname{SL}(2))$-comodules, among which the “transmutation” of $\mathcal{O}_{{q^2}}(\operatorname{SL}(2))$. 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 $q\to 1$ as regular functions over a suitable version of moduli spaces of twisted bundles.