## Stability of $\imath$canonical bases of irreducible finite type of real rank one

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- by Hideya Watanabe PDF
- Represent. Theory
**27**(2023), 1-29 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.

## Additional Information

**Hideya Watanabe**- Affiliation: Osaka Central Advanced Mathematical Institute, Osaka Metropolitan University, Osaka 558-8585, Japan
- MR Author ID: 1196919
- ORCID: 0000-0002-7705-8783
- Email: watanabehideya@gmail.com
- Received by editor(s): July 17, 2022
- Received by editor(s) in revised form: November 12, 2022, and December 19, 2022
- Published electronically: March 6, 2023
- Additional Notes: This work was supported by JSPS KAKENHI Grant Numbers JP20K14286 and JP21J00013.
- © Copyright 2023 American Mathematical Society
- Journal: Represent. Theory
**27**(2023), 1-29 - MSC (2020): Primary 17B37; Secondary 17B10
- DOI: https://doi.org/10.1090/ert/639