Miraculous cancellations for quantum ${\mathrm{SL}}_2$

Bonahon, Francis¹

¹ Department of Mathematics, University of Southern California, Los Angeles CA 90089-2532, U.S.A.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Volume spécial en l’honneur de Jean-Pierre OTAL “Low dimensional topology, hyperbolic manifolds and spectral geometry”, Tome 28 (2019) no. 3, pp. 523-557.

Abstract (VO)

In earlier work, Helen Wong and the author discovered certain “miraculous cancellations” for the quantum trace map connecting the Kauffman bracket skein algebra of a surface to its quantum Teichmüller space, occurring when the quantum parameter $q={\mathrm{e}}^{2\pi \mathrm{i}\hslash }$ is a root of unity. The current paper is devoted to giving a more representation theoretic interpretation of this phenomenon, in terms of the quantum group ${\mathrm{U}}_q\left({\mathrm{𝔰𝔩}}_2\right)$ and its dual Hopf algebra ${\mathrm{SL}}_2^q$.

Résumé

Des travaux précédents de Helen Wong et de l’auteur ont mis en évidence, quand le paramètre quantique $q={\mathrm{e}}^{2\pi \mathrm{i}\hslash }$ est une racine de l’unité, des « annulations miraculeuses » pour l’application de trace quantique qui relie l’algèbre d’écheveaux du crochet de Kauffman à l’espace de Teichmüller quantique d’une surface. L’article ci-dessous fournit une interprétation plus conceptuelle de ce phénomène, en termes de représentations du groupe quantique ${\mathrm{U}}_q\left({\mathrm{𝔰𝔩}}_2\right)$ et de son algèbre de Hopf duale ${\mathrm{SL}}_2^q$.

Détail

Publié le : 2019-12-06

DOI : 10.5802/afst.1608

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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