Stated skein algebras of surfaces
| Autor: | Costantino, Francesco, Le, Thang T. Q. |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0017,Quantact,Topologie quantique et géométrie de contact(2016), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011) |
| Rok vydání: | 2022 |
| Předmět: |
Mathematics - Geometric Topology
[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] Mathematics::Category Theory Mathematics::Quantum Algebra Applied Mathematics General Mathematics FOS: Mathematics Geometric Topology (math.GT) Mathematics::Geometric Topology Primary 57N10 secondary 57M25 |
| Zdroj: | Journal of the European Mathematical Society. 24:4063-4142 |
| ISSN: | 1435-9855 |
| Popis: | 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 ${\mathcal O}_{q^2}(\mathrm{SL}(2))$ 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 ${\mathcal O}_{q^2}(\mathrm{SL}(2))$ in a topological way. We deduce that the skein algebra of a surface with $n$ boundary components is an algebra-comodule over ${\mathcal O}_{q^2}(\mathrm{SL}(2))^{\otimes{n}}$ 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 "glueing on a triangle"; then we recover topologically some braided bialgebras in the category of ${\mathcal O}_{q^2}(\mathrm{SL}(2))$-comodules, among which the "transmutation" of ${\mathcal O}_{q^2}(\mathrm{SL}(2))$. 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. Comment: 74 pages, 33 figures. In version 2 : strengthened Theorem 4.17. To be published in the Journal of the European Mathematical Society |
| Databáze: | OpenAIRE |
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