Hostname: page-component-5b777bbd6c-cp4x8 Total loading time: 0 Render date: 2025-06-19T23:09:58.693Z Has data issue: false hasContentIssue false
  • English
  • Français

INTEGRABLE MODULES OVER QUANTUM SYMMETRIC PAIR COIDEAL SUBALGEBRAS

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  05 June 2025

Hideya Watanabe Open the ORCID record for Hideya Watanabe [Opens in a new window]
Hideya Watanabe*
Affiliation:
College of Science, https://ror.org/00x194q47 Rikkyo University , 3-34-1, Nishi-Ikebukuro, Toshima-ku, Tokyo, 171-8501, Japan
*
(watanabehideya@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce the notion of integrable modules over $\imath $quantum groups (a.k.a. quantum symmetric pair coideal subalgebras). After determining a presentation of such modules, we prove that each integrable module over a quantum group is integrable when restricted to an $\imath $quantum group. As an application, we show that the space of matrix coefficients of all simple integrable modules over an $\imath $quantum group of finite type with specific parameters coincides with Bao-Song’s coordinate ring of the $\imath $quantum group.

Keywords

quantum symmetric pairrepresentation theoryintegrable modulecanonical basis

MSC classification

Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 26
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bao, H and Song, J (2022) Symmetric subgroup schemes, Frobenius splittings, and quantum symmetric pairs. arXiv preprint 2212.13426.Google Scholar
Bao, H and Wang, W (2018) Canonical bases arising from quantum symmetric pairs. Invent. Math. 213(3), 1099–1177.10.1007/s00222-018-0801-5CrossRefGoogle Scholar
Bao, H and Wang, W (2021) Canonical bases arising from quantum symmetric pairs of Kac-Moody type. Compos. Math. 157(7), 1507–1537.10.1112/S0010437X2100734XCrossRefGoogle Scholar
Dijkhuizen, MS (1996) Some remarks on the construction of quantum symmetric spaces, Representations of Lie groups, Lie algebras and their quantum analogues. Acta Appl. Math. 44(1–2), 59–80.10.1007/BF00116516CrossRefGoogle Scholar
Drinfeld, VG (1985) Hopf algebras and the quantum Yang-Baxter equation. Dokl. Akad. Nauk SSSR 283(5), 1060–1064.Google Scholar
Gavrilik, AM and Klimyk, AU (1991) $q$ -deformed orthogonal and pseudo-orthogonal algebras and their representations. Lett. Math. Phys. 21(3), 215–220.10.1007/BF00420371CrossRefGoogle Scholar
Iorgov, N and Klimyk, AU (2005) Classification theorem on irreducible representations of the q-deformed algebra ${U}_q^{\prime}\left(\mathfrak{s}{\mathfrak{o}}_{\mathfrak{n}}\right)$ . Int. J. Math. Math. Sci. 2, 225–262.10.1155/IJMMS.2005.225CrossRefGoogle Scholar
Ito, T and Terwilliger, P (2010) The augmented tridiagonal algebra. Kyushu J. Math. 64(1), 81–144.10.2206/kyushujm.64.81CrossRefGoogle Scholar
Jimbo, M (1985) A $q$ -difference analogue of $U\left(\mathfrak{g}\right)$ and the Yang-Baxter equation. Lett. Math. Phys. 10(1), 63–69.10.1007/BF00704588CrossRefGoogle Scholar
Kashiwara, M (1993) Global crystal bases of quantum groups. Duke Math. J. 69(2), 455–485.10.1215/S0012-7094-93-06920-7CrossRefGoogle Scholar
Kolb, S (2014) Quantum symmetric Kac-Moody pairs. Adv. Math. 267, 395–469.10.1016/j.aim.2014.08.010CrossRefGoogle Scholar
Kolb, S and Stephens, J (2024) Representation theory of very non-standard quantum $\mathfrak{so}\left(2N-1\right)$ . arXiv preprint 2407.15538.Google Scholar
Letzter, G (1999) Symmetric pairs for quantized enveloping algebras. J. Algebra 220(2), 729–767.10.1006/jabr.1999.8015CrossRefGoogle Scholar
Letzter, G (2019) Cartan subalgebras for quantum symmetric pair coideals. Represent. Theory 23, 88–153.10.1090/ert/523CrossRefGoogle Scholar
Lusztig, G (1992) Canonical bases in tensor products. Proc. Nat. Acad. Sci. U.S.A. 89(17), 8177–8179.10.1073/pnas.89.17.8177CrossRefGoogle ScholarPubMed
Lusztig, G (2010) Introduction to Quantum Groups. Reprint of the 1994 edition. Modern Birkhäuser Classics. New York: Birkhäuser/Springer.Google Scholar
Molev, AI (2006) Representations of the twisted quantized enveloping algebra of type ${C}_n$ . Mosc. Math. J. 6(3), 531–551, 588.10.17323/1609-4514-2006-6-3-531-551CrossRefGoogle Scholar
Noumi, M (1996) Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. Adv. Math. 123(1), 16–77.10.1006/aima.1996.0066CrossRefGoogle Scholar
Noumi, M and Sugitani, T (1995) Quantum symmetric spaces and related $q$ -orthogonal polynomials. In Group Theoretical Methods in Physics (Toyonaka, 1994). River Edge, NJ: World Scientific Publishing Co., Inc, 28–40.Google Scholar
Procesi, C (2007) Lie Groups. An Approach Through Invariants and Representations. New York: Universitext, Springer.Google Scholar
Regelskis, V and Vlaar, B (2020) Quasitriangular coideal subalgebras of ${U}_q\left(\mathfrak{g}\right)$ in terms of generalized Satake diagrams. Bull. Lond. Math. Soc. 52(4), 693–715.10.1112/blms.12360CrossRefGoogle Scholar
Watanabe, H (2020) Crystal basis theory for a quantum symmetric pair. Int. Math. Res. Not. IMRN 22, 8292–8352.Google Scholar
Watanabe, H (2021) Classical weight modules over quantum groups. J. Algebra 578, 241–302.10.1016/j.jalgebra.2021.02.023CrossRefGoogle Scholar
Watanabe, H (2023) Based modules over the quantum group of type AI. Math. Z. 303(2), Paper No. 43, 73 pp.10.1007/s00209-022-03189-zCrossRefGoogle Scholar
Watanabe, H (2023) Stability of canonical bases of irreducible finite type of real rank one. Represent. Theory 27, 1–29.10.1090/ert/639CrossRefGoogle Scholar
Watanabe, H (2024) Stability of canonical bases of locally finite type. In Transformation Groups, https://doi.org/10.1007/s00031-024-09876-x.CrossRefGoogle Scholar
Wenzl, H (2020) On representations of ${U}_q^{\prime }{\mathfrak{so}}_n$ . Trans. Amer. Math. Soc. 373(5), 3295–3322.10.1090/tran/7983CrossRefGoogle Scholar