Stability of 𝚤canonical bases of irreducible finite type of real rank one

Watanabe, Hideya

doi:10.1090/ert/639

Stability of $\imath$canonical bases of irreducible finite type of real rank one
HTML articles powered by AMS MathViewer

by Hideya Watanabe;

Represent. Theory 27 (2023), 1-29

DOI: https://doi.org/10.1090/ert/639

Published electronically: March 6, 2023

HTML | PDF | Request permission

Abstract:

It has been known since their birth in Bao and Wang’s work that the $\imath$canonical bases of $\imath$quantum groups are not stable in general. In the author’s previous work, the stability of $\imath$canonical bases of certain quasi-split types turned out to be closely related to the theory of $\imath$crystals. In this paper, we prove the stability of $\imath$canonical bases of irreducible finite type of real rank $1$, for which the theory of $\imath$crystals has not been developed, by means of global and local crystal bases.

References

Andrea Appel and Bart Vlaar, Universal K-matrices for quantum Kac-Moody algebras, Represent. Theory 26 (2022), 764–824. MR 4454332, DOI 10.1090/ert/623
Shôrô Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ. 13 (1962), 1–34. MR 153782
Martina Balagović and Stefan Kolb, The bar involution for quantum symmetric pairs, Represent. Theory 19 (2015), 186–210. MR 3414769, DOI 10.1090/ert/469
Martina Balagović and Stefan Kolb, Universal K-matrix for quantum symmetric pairs, J. Reine Angew. Math. 747 (2019), 299–353. MR 3905136, DOI 10.1515/crelle-2016-0012
Huanchen Bao and Weiqiang Wang, Canonical bases in tensor products revisited, Amer. J. Math. 138 (2016), no. 6, 1731–1738. MR 3595499, DOI 10.1353/ajm.2016.0051
Huanchen Bao and Weiqiang Wang, Canonical bases arising from quantum symmetric pairs, Invent. Math. 213 (2018), no. 3, 1099–1177. MR 3842062, DOI 10.1007/s00222-018-0801-5
Huanchen Bao and Weiqiang Wang, A new approach to Kazhdan-Lusztig theory of type $B$ via quantum symmetric pairs, Astérisque 402 (2018), vii+134 (English, with English and French summaries). MR 3864017
Huanchen Bao and Weiqiang Wang, Canonical bases arising from quantum symmetric pairs of Kac-Moody type, Compos. Math. 157 (2021), no. 7, 1507–1537. MR 4277109, DOI 10.1112/S0010437X2100734X
Daniel Bump and Anne Schilling, Crystal bases, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. Representations and combinatorics. MR 3642318, DOI 10.1142/9876
V. G. Drinfel′d, Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060–1064 (Russian). MR 802128
Michael Ehrig and Catharina Stroppel, Nazarov-Wenzl algebras, coideal subalgebras and categorified skew Howe duality, Adv. Math. 331 (2018), 58–142. MR 3804673, DOI 10.1016/j.aim.2018.01.013
The GAP Group, GAP – groups, algorithms, and programming, Version 4.11.1, 2021, https://www.gap-system.org.
W. A. de Graaf and T. GAP Team, QuaGroup, Computations with quantum groups, Version 1.8.3, Refereed GAP package, 2022, https://gap-packages.github.io/quagroup/.
Michio Jimbo, A $q$-difference analogue of $U({\mathfrak {g}})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001, DOI 10.1007/BF00704588
Michio Jimbo, A $q$-analogue of $U({\mathfrak {g}}{\mathfrak {l}}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252. MR 841713, DOI 10.1007/BF00400222
M. Kashiwara, On crystal bases of the $Q$-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516. MR 1115118, DOI 10.1215/S0012-7094-91-06321-0
Masaki Kashiwara, The crystal base and Littelmann’s refined Demazure character formula, Duke Math. J. 71 (1993), no. 3, 839–858. MR 1240605, DOI 10.1215/S0012-7094-93-07131-1
Masaki Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383–413. MR 1262212, DOI 10.1215/S0012-7094-94-07317-1
Stefan Kolb, Quantum symmetric Kac-Moody pairs, Adv. Math. 267 (2014), 395–469. MR 3269184, DOI 10.1016/j.aim.2014.08.010
Stefan Kolb, The bar involution for quantum symmetric pairs—hidden in plain sight, Hypergeometry, integrability and Lie theory, Contemp. Math., vol. 780, Amer. Math. Soc., [Providence], RI, [2022] ©2022, pp. 69–77. MR 4476416, DOI 10.1090/conm/780/15687
Gail Letzter, Symmetric pairs for quantized enveloping algebras, J. Algebra 220 (1999), no. 2, 729–767. MR 1717368, DOI 10.1006/jabr.1999.8015
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365–421. MR 1088333, DOI 10.1090/S0894-0347-1991-1088333-2
G. Lusztig, Canonical bases in tensor products, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 17, 8177–8179. MR 1180036, DOI 10.1073/pnas.89.17.8177
George Lusztig, Introduction to quantum groups, Modern Birkhäuser Classics, Birkhäuser/Springer, New York, 2010. Reprint of the 1994 edition. MR 2759715, DOI 10.1007/978-0-8176-4717-9
Vidas Regelskis and Bart Vlaar, Quasitriangular coideal subalgebras of $U_q(\mathfrak {g})$ in terms of generalized Satake diagrams, Bull. Lond. Math. Soc. 52 (2020), no. 4, 693–715. MR 4171396, DOI 10.1112/blms.12360
H. Watanabe, Based modules over the $\imath$quantum group of type AI, Math. Z., To appear, arXiv:2103.12932.
H. Watanabe, Crystal bases of modified $\imath$quantum groups of certain quasi-split types, arXiv:2110.07177, 2021.

AMS Website Down

Representation Theory

Stability of $\imath$canonical bases of irreducible finite type of real rank one
HTML articles powered by AMS MathViewer

Abstract: