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

Francis Bonahon

doi:10.5802/afst.1608

Miraculous cancellations for quantum

{SL}_{2}

Francis Bonahon ¹

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)
Résumé

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 $q = e^{2 π i ℏ}$ 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 $U_{q} ({𝔰𝔩}_{2})$ and its dual Hopf algebra ${SL}_{2}^{q}$ .

Des travaux précédents de Helen Wong et de l’auteur ont mis en évidence, quand le paramètre quantique $q = e^{2 π i ℏ}$ 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 $U_{q} ({𝔰𝔩}_{2})$ et de son algèbre de Hopf duale ${SL}_{2}^{q}$ .

Publié le : 2019-12-06

DOI : 10.5802/afst.1608

Affiliations des auteurs :

Francis Bonahon ¹

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Francis Bonahon. Miraculous cancellations for quantum $\protect \mathrm{SL}_2$. 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. doi: 10.5802/afst.1608

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