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

Bonahon, Francis

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 3, pp. 523-557.

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 ${\mathrm{SL}}_2^q$.

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 ${\mathrm{SL}}_2^q$.

[1] Bonahon, Francis; Otal, Jean-Pierre Scindements de Heegaard des espaces lenticulaires, C. R. Math. Acad. Sci. Paris, Volume 294 (1982) no. 17, pp. 585-587 | MR | Zbl

[2] Bonahon, Francis; Otal, Jean-Pierre Scindements de Heegaard des espaces lenticulaires, Ann. Sci. Éc. Norm. Supér., Volume 16 (1983) no. 3, pp. 451-466 | DOI | MR | Zbl

[3] Bonahon, Francis; Otal, Jean-Pierre Variétés hyperboliques à géodésiques arbitrairement courtes, Bull. Lond. Math. Soc., Volume 20 (1988) no. 3, pp. 255-261 | DOI | MR | Zbl

[4] Bonahon, Francis; Otal, Jean-Pierre Laminations mesurées de plissage des variétés hyperboliques de dimension $3$, Ann. Math., Volume 160 (2004) no. 3, pp. 1013-1055 | DOI | MR | Zbl

[5] Bonahon, Francis; Wong, Helen Quantum traces for representations of surface groups in ${\mathrm{SL}}_2\left(ℂ\right)$, Geom. Topol., Volume 15 (2011) no. 3, pp. 1569-1615 | DOI | MR | Zbl

[6] Bonahon, Francis; Wong, Helen Representations of the Kauffman bracket skein algebra I: invariants and miraculous cancellations, Invent. Math., Volume 204 (2016) no. 1, pp. 195-243 | DOI | MR | Zbl

[7] Drinfeld, Vladimir G. Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR, Volume 283 (1985) no. 5, pp. 1060-1064 | MR

[8] Drinfeld, Vladimir G. Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, 1986), American Mathematical Society (1987), pp. 798-820 | MR

[9] Garoufalidis, Stavros; Lê, Thang T. Q.; Zeilberger, Doron The quantum MacMahon master theorem, Proc. Natl. Acad. Sci. USA, Volume 103 (2006) no. 38, pp. 13928-13931 | DOI | MR | Zbl

[10] Jimbo, Michio A $q$-difference analogue of $\mathrm{U}\left(𝔤\right)$ and the Yang-Baxter equation, Lett. Math. Phys., Volume 10 (1985) no. 1, pp. 63-69 | DOI | MR | Zbl

[11] Kassel, Christian Quantum groups, Graduate Texts in Mathematics, 155, Springer, 1995, xii+531 pages | DOI | MR | Zbl

[12] Kirillov, Anatoliĭ N.; Reshetikhin, Nicolai Yu. Representations of the algebra ${\mathrm{U}}_q\left(\mathrm{sl}\left(2\right)\right),q$-orthogonal polynomials and invariants of links, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988) (Advanced Series in Mathematical Physics), Volume 7, World Scientific, 1989, pp. 285-339 | MR

[13] Koelink, H. T.; Koornwinder, Tom H. The Clebsch–Gordan coefficients for the quantum group ${\mathrm{S}}_{\mu }\mathrm{U}\left(2\right)$ and $q$-Hahn polynomials, Indag. Math., Volume 51 (1989) no. 4, pp. 443-456 | DOI | MR | Zbl

[14] Manin, Yuri I. Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier, Volume 37 (1987) no. 4, pp. 191-205 | DOI | MR | Zbl

[15] Manin, Yuri I. Quantum groups and noncommutative geometry, Université de Montréal, 1988, vi+91 pages | MR | Zbl

[16] Reshetikhin, Nicolai Yu.; Takhtadzhyan, Leon A.; Faddeev, Lyudvig D. Quantization of Lie groups and Lie algebras, Algebra Anal., Volume 1 (1989) no. 1, pp. 178-206 | MR

[17] Reshetikhin, Nicolai Yu.; Turaev, Vladimir G. Ribbon graphs and their invariants derived from quantum groups, Commun. Math. Phys., Volume 127 (1990) no. 1, pp. 1-26 | DOI | MR | Zbl

[18] Reshetikhin, Nicolai Yu.; Turaev, Vladimir G. Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math., Volume 103 (1991) no. 3, pp. 547-597 | DOI | MR | Zbl

[19] Takeuchi, Mitsuhiro Hopf algebra techniques applied to the quantum group ${\mathrm{U}}_q\left(𝔰l\left(2\right)\right)$, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990) (Contemporary Mathematics), Volume 134, American Mathematical Society, 1992, pp. 309-323 | DOI | MR | Zbl

[20] Takeuchi, Mitsuhiro Some topics on ${\mathrm{GL}}_q\left(n\right)$, J. Algebra, Volume 147 (1992) no. 2, pp. 379-410 | DOI | MR | Zbl

[21] Vaksman, Leonid L. $q$-analogues of Clebsch-Gordan coefficients, and the algebra of functions on the quantum group $\mathrm{SU}\left(2\right)$, Dokl. Akad. Nauk SSSR, Volume 306 (1989) no. 2, pp. 269-271 | MR | Zbl

[22] Witten, Edward Quantum field theory and the Jones polynomial, Commun. Math. Phys., Volume 121 (1989) no. 3, pp. 351-399 | DOI | MR | Zbl