Classical shadows of stated skein representations at roots of unity

Korinman, Julien; Quesney, Alexandre

doi:10.2140/agt.2024.24.2091

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Classical shadows of stated skein representations at roots of unity

Julien Korinman and Alexandre Quesney

Algebraic & Geometric Topology 24 (2024) 2091–2148

DOI: 10.2140/agt.2024.24.2091

Abstract

We extend some results of Bonahon, Wong, Bullock and Turaev concerning the skein algebras of closed surfaces to Lê’s stated skein algebras associated to open surfaces. We prove that the stated skein algebra with deforming parameter $+ 1$ embeds canonically into the center of the stated skein algebra whose deforming parameter is an odd root unity. We also construct an isomorphism between the stated skein algebra at $+ 1$ and the algebra of regular functions of the relative ${SL}_{2}$ –character variety of the surface. As a result, we associate to each isomorphism class of irreducible or local representations of the stated skein algebra an invariant which is a point in the relative character variety.

Keywords

stated skein algebras, character varieties

Mathematical Subject Classification

Primary: 57R56

References

Publication

Received: 16 March 2022

Revised: 25 March 2023

Accepted: 24 April 2023

Published: 16 July 2024

Authors

Julien Korinman
Institut Montpelliérain Alexandre Grothendieck Université de Montpellier Montpellier France
https://sites.google.com/site/homepagejulienkorinman/
Alexandre Quesney
ETS Ingenieros Informáticos Universidad Politécnica de Madrid Campus de Montegancedo Madrid Spain

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Volume 24, issue 4 (2024)