Geodesic
Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik–Zamolodchikov equations and Letzter–Kolb coideals
Forum of Mathematics, Pi, Tome 11 (2023)

Voir la notice de l'article provenant de la source Cambridge University Press

We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingof cyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno–Drinfeld type theorem on type $\mathrm {B}$ braid group representations defined by the monodromy of KZ-equations and by the Balagović–Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions. 
@article{10_1017_fmp_2023_11,
     author = {Kenny De Commer and Sergey Neshveyev and Lars Tuset and Makoto Yamashita},
     title = {Comparison of quantizations of symmetric spaces: cyclotomic {Knizhnik{\textendash}Zamolodchikov} equations and {Letzter{\textendash}Kolb} coideals},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {11},
     year = {2023},
     doi = {10.1017/fmp.2023.11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.11/}
}
TY  - JOUR
AU  - Kenny De Commer
AU  - Sergey Neshveyev
AU  - Lars Tuset
AU  - Makoto Yamashita
TI  - Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik–Zamolodchikov equations and Letzter–Kolb coideals
JO  - Forum of Mathematics, Pi
PY  - 2023
VL  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.11/
DO  - 10.1017/fmp.2023.11
LA  - en
ID  - 10_1017_fmp_2023_11
ER  - 
%0 Journal Article
%A Kenny De Commer
%A Sergey Neshveyev
%A Lars Tuset
%A Makoto Yamashita
%T Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik–Zamolodchikov equations and Letzter–Kolb coideals
%J Forum of Mathematics, Pi
%D 2023
%V 11
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2023.11/
%R 10.1017/fmp.2023.11
%G en
%F 10_1017_fmp_2023_11
Kenny De Commer; Sergey Neshveyev; Lars Tuset; Makoto Yamashita. Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik–Zamolodchikov equations and Letzter–Kolb coideals. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.11

[Ara62] Araki, S., ‘On root systems and an infinitesimal classification of irreducible symmetric spaces’, J. Math. Osaka City Univ. 13 (1962), 1–34. MR0153782.Google Scholar

[BK15] Balagović, M. and Kolb, S., ‘The bar involution for quantum symmetric pairs’, Represent. Theory 19 (2015), 186–210, DOI:10.1090/ert/469. MR3414769.Google Scholar | DOI

[BK19] Balagović, M. and Kolb, S., ‘Universal K-matrix for quantum symmetric pairs’, J. Reine Angew. Math. 747 (2019), 299–353. DOI:10.1515/crelle-2016-0012. MR3905136.Google Scholar | DOI

[BW18] Bao, H. and Wang, W., ‘A new approach to Kazhdan–Lusztig theory of type via quantum symmetric pairs’, Astérisque 402 (2018), vii+134. MR3864017.Google Scholar

[BGI71] Berthelot, P., Grothendieck, A. and L, Illusie (eds.) Théorie des intersections et théorème de Riemann–Roch, Lecture Notes in Mathematics, vol. 225 (Springer-Verlag, Berlin-New York, 1971) (French). Séminaire de Géométrie Algébrique du Bois-Marie 1966–1967 (SGA 6); Avec la collaboration de Ferrand, D., Jouanolou, J. P., Jussila, O., Kleiman, S., Raynaud, M. et J. P. Serre. MR0354655.Google Scholar | DOI

[Bor98] Borel, A., ‘Semisimple Groups and Riemannian Symmetric Spaces’, Texts and Readings in Mathematics, vol. 16 (Hindustan Book Agency, New Delhi, 1998). MR1661166.Google Scholar | DOI

[Bou07] Bourbaki, N., ‘Éléments de mathématique’, Groupes et algèbres de Lie . chapitre 7 et 8, (Springer, Berlin-Heidelberg, 2007). Reprint of the 1975 (Hermann) edition.Google Scholar

[Bro12] Adrien, B., ‘A Kohno–Drinfeld theorem for the monodromy of cyclotomic KZ connections’, Comm. Math. Phys. 311(1) (2012), 55–96, DOI:10.1007/s00220-012-1424-0. MR2892463.Google Scholar

[Bro13] Brochier, A., ‘Cyclotomic associators and finite type invariants for tangles in the solid torus’, Algebr. Geom. Topol. 13(6) (2013), 3365–3409. DOI:10.2140/agt.2013.13.3365. MR3248737.Google Scholar | DOI

[Cal06] Calaque, D., ‘Quantization of formal classical dynamical -matrices: the reductive case’, Adv. Math. 204(1) (2006), 84–100. DOI:10.1016/j.aim.2005.05.009. MR2233127.Google Scholar | DOI

[Che84] Cherednik, I. V., ‘Factorizing particles on a half line, and root systems’, Teoret. Mat. Fiz. 61(1) (1984), 35–44. MR774205.Google Scholar

[DCM20] De Commer, K. and Matassa, M., ‘Quantum flag manifolds, quantum symmetric spaces and their associated universal -matrices’, Adv. Math. 366 (2020), 107029, 100. DOI:10.1016/j.aim.2020.107029, [math.QA]. MR4070299.Google Scholar | arXiv | DOI

[DCNTY19] De Commer, K., Neshveyev, S., Tuset, L. and Yamashita, M., ‘Ribbon braided module categories, quantum symmetric pairs and Knizhnik–Zamolodchikov equations’, Comm. Math. Phys. 367(3) (2019), 717–769. DOI:10.1007/s00220-019-03317-7. MR3943480.Google Scholar | DOI

[DCY13] De Commer, K. and Yamashita, M., ‘Tannaka–Kreĭn duality for compact quantum homogeneous spaces. I. General theory’, Theory Appl. Categ. 28(31) (2013), 1099–1138.Google Scholar

[tDHO98] Tom Dieck, T. and Häring-Oldenburg, R., ‘Quantum groups and cylinder braiding’, Forum Math. 10(5) (1998), 619–639. DOI:10.1515/form.10.5.619. MR1644317.Google Scholar | DOI

[Dix55] Dixmier, J., ‘Cohomologie des algèbres de Lie nilpotentes’, Acta Sci. Math. Szeged 16 (1955), 246–250. MR0074780.Google Scholar

[Dol05] Dolgushev, V., ‘Covariant and equivariant formality theorems’, Adv. Math. 191(1) (2005), 147–177. DOI:10.1016/j.aim.2004.02.001. MR102846 (2006c:53101).Google Scholar | DOI

[DG95] Donin, J. and Gurevich, D., ‘Some Poisson structures associated to Drinfeld–Jimbo -matrices and their quantization’, Israel J. Math. 92(1–3) (1995), 23–32. DOI:10.1007/BF02762068. MR1357743.Google Scholar | DOI

[DGS99] Donin, J., Gurevich, D. and Shnider, S., ‘Double quantization on some orbits in the coadjoint representations of simple Lie groups’, Comm. Math. Phys. 204(1) (1999), 39–60. DOI:10.1007/s002200050636. MR1705663.Google Scholar | DOI

[DM03a] Donin, J. and Mudrov, A., ‘Reflection equation, twist, and equivariant quantization’, Israel J. Math. 136 (2003), 11–28. DOI:10.1007/BF02807191. MR1998103.Google Scholar | DOI

[DM03b] Donin, J. and Mudrov, A., ‘Method of quantum characters in equivariant quantization’, Comm. Math. Phys. 234(3) (2003), 533–555. DOI:10.1007/s00220-002-0771-7. MR1964382.Google Scholar | DOI

[DS97] Donin, J. and Shnider, S., ‘Cohomological construction of quantized universal enveloping algebras’, Trans. Amer. Math. Soc. 349(4) (1997), 1611–1632. DOI:10.1090/S0002-9947-97-01787-X. MR1390978.Google Scholar | DOI

[Dri89a] Drinfel’D, V. G., ‘On almost cocommutative Hopf algebras’, Algebra i Analiz 1(2) (1989), 30–46. Translation in Leningrad Math. J. 1(2) (1990), 321–342. MR1025154 (91b:16046).Google Scholar

[Dri89b] Drinfeld, V. G., ‘Quasi-Hopf algebras’, Algebra i Analiz 1(6) (1989), 114–148. Translation in Leningrad Math. J. (6) (1990), 1419–1457. MR1047964 (91b:17016).Google Scholar

[Dri90] Drinfeld, V. G., ‘On quasitriangular quasi-Hopf algebras and on a group that is closely connected with ’, Algebra i Analiz 2(4) (1990), 149–181. (Russian); English transl., Leningrad Math. J. (4) (1991), 829–860. MR1080203.Google Scholar

[Dri93] Drinfeld, V. G., ‘On Poisson homogeneous spaces of Poisson–Lie groups’, Teoret. Mat. Fiz. 95(2) (1993), 226–227. MR1243249 (94k:58045).Google Scholar

[Enr07] Enriquez, B., ‘Quasi-reflection algebras and cyclotomic associators’, Selecta Math. (N.S.) 13(3) (2007), 391–463. DOI:10.1007/s00029-007-0048-2. MR2383601 (2010e:17009).Google Scholar | DOI

[EE05] Enriquez, B. and Etingof, P., ‘Quantization of classical dynamical -matrices with nonabelian base’, Comm. Math. Phys. 254(3) (2005), 603–650. DOI:10.1007/s00220-004-1243-z. MR2126485 (2006a:17011).Google Scholar | DOI

[EV98] Etingof, P. and Varchenko, A., ‘Geometry and classification of solutions of the classical dynamical Yang–Baxter equation’, Comm. Math. Phys. 192(1)(1998), 77–120. DOI:10.1007/s002200050292. MR1612160.Google Scholar | DOI

[Fel95] Felder, G., Conformal Field Theory and Integrable Systems Associated to Elliptic Curves, Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Birkhäuser, Basel, Zürich, 1994; 1995), 1247–1255. MR1404026 (97m:81015).Google Scholar

[FL04] Foth, P. and Lu, J.-H., ‘A Poisson structure on compact symmetric space’s’, Comm. Math. Phys. 251(3) (2004), 557–566. DOI:10.1007/s00220-004-1178-4. MR2102330.Google Scholar | DOI

[GL00] Golubeva, V. A., and Leksin, V., On a generalization of the Drinfeld-Kohno theorem, Proceedings of the Second ISAAC Congress, Vol. 2 (Fukuoka, 1999; Kluwer Acad. Publ., Dordrecht, 2000), 1371–1386. MR1940926 (2004a:32023).Google Scholar

[GS99] Gurevich, D. and Saponov, P., ‘Quantum sphere via reflection equation algebra’, Preprint, 1999, .Google Scholar | arXiv

[HC55] Harish-Chandra, , ‘Representations of semisimple Lie groups. IV’, Amer. J. Math. 77 (1955), 743–777, DOI:10.2307/2372596. MR0072427.Google Scholar | DOI

[HC56] Harish-Chandra, , ‘Representations of semisimple Lie groups. VI. Integrable and square-integrable representations’, Amer. J. Math. 78 (1956), 564–628, DOI:10.2307/2372674. MR0082056.Google Scholar | DOI

[Hel01] Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics, vol. 34 (American Mathematical Society, Providence, RI, 2001). DOI:10.1090/gsm/034, ISBN 0-8218-2848-7. Corrected reprint of the 1978 original. MR1834454.Google Scholar

[Jan96] Jantzen, J. C., Lectures on Quantum Groups, Graduate Studies in Mathematics, vol. 6, (American Mathematical Society, Providence, RI, 1996). MR1359532.Google Scholar

[Kac90] Kac, V. G., Infinite-Dimensional Lie Algebras, third edn. (Cambridge University Press, Cambridge,1990). DOI:10.1017/CBO9780511626234, MR1104219 (92k:17038).Google Scholar | DOI

[KW92] Kac, V. G. and Wang, S. P., ‘On automorphisms of Kac–Moody algebras and groups’, Adv. Math. 92(2) (1992), 129–195. MR1155464.Google Scholar | DOI

[Kar96] Karolinskiĭ, E. A., ‘Classification of Poisson homogeneous spaces of a compact Poisson–Lie group’, Mat. Fiz. Anal. Geom. 3(3–4) (1996), 274–289. MR1485682.Google Scholar

[Kas95] Kassel, C., Quantum Groups, Graduate Texts in Mathematics, vol. 155 (Springer-Verlag, New York, 1995). MR1321145 (96e:17041).Google Scholar | DOI

[Kna02] Knapp, A. W., Lie Groups Beyond an Introduction , second edn., Progress in Mathematics, vol. 140, (Birkhäuser Boston, Inc., Boston, MA, 2002). MR1920389.Google Scholar

[Koh87] Kohno, T., ‘Monodromy representations of braid groups and Yang–Baxter equations’, Ann. Inst. Fourier (Grenoble) 37(4) (1987), 139–160. MR927394 (89h:17030).Google Scholar | DOI

[Kol08] Kolb, S., ‘Quantum symmetric pairs and the reflection equation’, Algebr. Represent. Theory 11(6) (2008), 519–544. DOI:10.1007/s10468-008-9093-6. MR2453228.Google Scholar | DOI

[Kol14] Kolb, S., ‘Quantum symmetric Kac–Moody pairs’, Adv. Math. 267 (2014), 395–469. DOI:10.1016/j.aim.2014.08.010. MR3269184.Google Scholar | DOI

[Kol20] Kolb, S., ‘Braided module categories via quantum symmetric pairs’, Proc. Lond. Math. Soc. (3) 121(1) (2020), 1–31. DOI:10.1112/plms.12303. MR4048733.Google Scholar | DOI

[KS09] Kolb, S. and Stokman, J. V., ‘Reflection equation algebras, coideal subalgebras, and their centres’, Selecta Math. (N.S.) 15(4) (2009), 621–664. DOI:10.1007/s00029-009-0007-1. MR2565052.Google Scholar | DOI

[Koo93] Koornwinder, T. H., ‘Askey–Wilson polynomials as zonal spherical functions on the quantum group’, SIAM J. Math. Anal. 24(3) (1993), 795–813. MR1215439 (94k:33042).Google Scholar | DOI

[Lei94] Leibman, A., ‘Some monodromy representations of generalized braid groups’, Comm. Math. Phys. 164(2) (1994), 293–304. MR1289327 (95k:52019).Google Scholar | DOI

[Let99] Letzter, G., ‘Symmetric pairs for quantized enveloping algebras’, J. Algebra 220(2) (1999), 729–767. MR1717368.Google Scholar | DOI

[Let00] Letzter, G., ‘Harish–Chandra modules for quantum symmetric pairs’, Represent. Theory 4 (2000), 64–96. DOI:10.1090/S1088-4165-00-00087-X. MR1742961.Google Scholar | DOI

[LV12] Loday, J.-L. and Vallette, B., Algebraic Operads, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346 (Springer, Heidelberg, 2012). MR2954392.Google Scholar | DOI

[LW90] Lu, J.-H. and Weinstein, A., ‘Poisson Lie groups, dressing transformations, and Bruhat decompositions’, J. Differential Geom. 31(2) (1990), 501–526. MR1037412.Google Scholar | DOI

[Lus10] Lusztig, G., Introduction to Quantum Groups, Modern Birkhäuser Classics (Birkhäuser/Springer, New York, 2010). DOI:10.1007/978-0-8176-4717-9. Reprint of the 1994 edition. MR2759715.Google Scholar

[Moo64] Moore, C. C., ‘Compactifications of symmetric spaces. II. The Cartan domains’, Amer. J. Math. 86 (1964), 358–378. MR0161943.Google Scholar | DOI

[Mud02] Mudrov, A., ‘Characters of -reflection equation algebra’, Lett. Math. Phys. 60(3) (2002), 283–291. DOI:10.1023/A:1016283527111. MR1917138.Google Scholar | DOI

[Nes14] Neshveyev, S., ‘Duality theory for nonergodic actions’, Münster J. Math. 7(2) (2014), 413–437. MR3426224.Google Scholar

[NT11] Neshveyev, S. and Tuset, L., ‘Notes on the Kazhdan–Lusztig theorem on equivalence of the Drinfeld category and the category of -modules’, Algebr. Represent. Theory 14(5) (2011), 897–948. DOI:10.1007/s10468-010-9223-9. MR2832264 (2012j:17022).Google Scholar | DOI

[Nou96] Noumi, M., ‘Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces’, Adv. Math. 123(1) (1996), 16–77. DOI:10.1006/aima.1996.0066. MR1413836.Google Scholar | DOI

[OV90] Onishchik, A. L. and Vinberg, È. B., Lie Groups and Algebraic Groups, Springer Series in Soviet Mathematics (Springer-Verlag, Berlin, 1990). DOI:10.1007/978-3-642-74334-4. Translated from the Russian and with a preface by D. A. Leites. MR1064110.Google Scholar | DOI

[Ost03] Ostrik, V., ‘Module categories, weak Hopf algebras and modular invariants’, Transform. Groups 8(2) (2003), 177–206. DOI:10.1007/s00031-003-0515-6. MR1976459 (2004h:18006).Google Scholar | DOI

[Pod87] Podleś, P., ‘Quantum spheres’, Lett. Math. Phys. 14(3) (1987), 193–202. MR919322 (89b:46081).Google Scholar | DOI

[She91] Sheu, A. J.-L., ‘Quantization of the Poisson and its Poisson homogeneous space—The -sphere’, Comm. Math. Phys. 135(2) (1991), 217–232. With an appendix by Lu, Jiang-Hua and Weinstein, Alan. MR1087382 (91m:58011).Google Scholar | DOI

[Sto03] Stokman, J. V., ‘The quantum orbit method for generalized flag manifolds’, Math. Res. Lett. 10(4) (2003), 469–481. MR1995786 (2004e:20086).Google Scholar | DOI

[Tit66] Tits, J., ‘Sur les constantes de structure et le théorème d’existence des algèbres de Lie semi-simples’, Inst. Hautes Études Sci. Publ. Math. 31 (1966), 21–58. MR0214638.Google Scholar | DOI

[VD96] Van Daele, A., Discrete quantum groups, J. Algebra 180(2) (1996), 431–444. MR1378538 (97a:16076).Google Scholar | DOI

[Wor87] Woronowicz, S. L., ‘Twisted group. An example of a noncommutative differential calculus’, Publ. Res. Inst. Math. Sci. 23(1) (1987), 117–181. MR890482 (88h:46130).Google Scholar | DOI

[Zus08] Zusmanovich, P., ‘A converse to the second Whitehead lemma’, J. Lie Theory 18(2) (2008), 295–299. MR2431117.Google Scholar

Cité par Sources :