Stated skein algebras of surfaces

  • Francesco Costantino

    Université Paul Sabatier de Toulouse, France
  • Thang T. Q. Lê

    Georgia Institut of Technology, Atlanta, USA
Stated skein algebras of surfaces cover
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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 Oq2(SL(2))\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 RR-matrix for Oq2(SL(2)\mathcal{O}_{{q^2}}(\operatorname{SL}(2) in a topological way. We deduce that the skein algebra of a surface with nn boundary components is a comodule algebra over Oq2(SL(2))n\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 Oq2(SL(2))\mathcal{O}_{{q^2}}(\operatorname{SL}(2))-comodules, among which the “transmutation” of Oq2(SL(2))\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 q1q\to 1 as regular functions over a suitable version of moduli spaces of twisted bundles.

Cite this article

Francesco Costantino, Thang T. Q. Lê, Stated skein algebras of surfaces. J. Eur. Math. Soc. 24 (2022), no. 12, pp. 4063–4142

DOI 10.4171/JEMS/1167