Skip to main content
SpringerLink
Log in

Kirillov–Reshetikhin Modules and Quantum K-matrices

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

From a quantum K-matrix of a fundamental representation of any quantum affine algebra, we construct one for the Kirillov–Reshetikhin module by fusion construction. Using the \(\imath \)crystal theory by the last author, we also obtain combinatorial K-matrices corresponding to the symmetric tensor representations of affine type A for all quasi-split Satake diagrams.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Appel, A., Vlaar, B.: Universal \(K\)-matrices for quantum Kac–Moody algebras. Represent. Theory 26, 764–824 (2022)

    Article  MathSciNet  Google Scholar 

  2. Appel, A., Vlaar, B.: Trigonometric \(K\)-matrices for finite-dimensional representations of quantum affine algebras. arXiv:2203.16503

  3. Balagović, M., Kolb, S.: Universal K-matrix for quantum symmetric pairs. J. Reine Angew. Math. 747, 299–353 (2019)

    Article  MathSciNet  Google Scholar 

  4. Bao, H., Wang, W.: A new approach to Kazhdan–Lusztig theory of type \(B\) via quantum symmetric pairs. Astérisque (402), vii+134 pp (2018)

  5. Bao, H., Wang, W.: Canonical bases arising from quantum symmetric pairs. Invent. Math. 213(3), 1099–1177 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  6. Bao, H., Wang, W.: Canonical bases arising from quantum symmetric pairs of Kac–Moody type. Compos. Math. 157(7), 1507–1537 (2021)

    Article  MathSciNet  Google Scholar 

  7. Caudrelier, V., Crampé, N., Zhang, Q.: Set-theoretical reflection equation: classification of reflection maps. J. Phys. A 46(9), 095203 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  8. Cherednik, I.: Factorizing particles on a half line, and root systems. Teoret. Mat. Fiz. 61(1), 35–44 (1984)

    MathSciNet  Google Scholar 

  9. Date, E., Okado, M.: Calculation of excitation spectra of the spin model related with the vector representation of the quantized affine algebra of type \(A_n^{(1)}\). Int. J. Mod. Phys. A 9, 399–417 (1994)

    Article  ADS  Google Scholar 

  10. De Commer, K.: Actions of skew braces and set-theoretic solutions of the reflection equation. Proc. Edinb. Math. Soc. (2) 62(4), 1089–1113 (2019)

    Article  MathSciNet  Google Scholar 

  11. Doikou, A., Smoktunowicz, A.: Set-theoretic Yang–Baxter & reflection equations and quantum group symmetries. Lett. Math. Phys. 111(4), Paper No. 105 (2021)

  12. Fukuda, K., Okado, M., Yamada, Y.: Energy functions in box-ball systems. Int. J. Mod. Phys. A 15, 1379–1392 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  13. Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Tsuboi, Z.: Paths, crystals and fermionic formulae. In: MathPhys Odyssey 2001, Progress in Mathematical Physics, vol. 23, pp. 205–272. Birkhäuser, Boston (2002)

  14. Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Remarks on fermionic formula. Contemp. Math. 248, 243–291 (1999)

    Article  MathSciNet  Google Scholar 

  15. Hatayama, G., Kuniba, A., Okado, M., Takagi, T., Yamada, Y.: Scattering rules in soliton cellular automata associated with crystal bases. Contemp. Math. 297, 151–182 (2002)

    Article  MathSciNet  Google Scholar 

  16. Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  17. Kashiwara, M.: On crystal bases of the \(q\)-analogue of universal enveloping algebras. Duke Math. J. 63, 465–516 (1991)

    Article  MathSciNet  Google Scholar 

  18. Kashiwara, M.: On crystal bases. In: Representations of Groups (Banff, AB), CMS Conference Proceedings, vol. 16, pp. 155–197. American Mathematical Society, Providence (1995)

  19. Kashiwara, M.: On level zero representations of quantized affine algebras. Duke Math. J. 112, 117–175 (2002)

    Article  MathSciNet  Google Scholar 

  20. Kolb, S.: Quantum symmetric Kac–Moody pairs. Adv. Math. 267, 395–469 (2014)

    Article  MathSciNet  Google Scholar 

  21. Kulish, P.P., Sklyanin, E.K.: Algebraic structures related to reflection equations. J. Phys. A 25(22), 5963–5975 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  22. Kuniba, A., Okado, M., Yamada, Y.: Box-ball system with reflecting end. J. Nonlinear Math. Phys. 12, 475–507 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  23. Kuniba, A., Okado, M., Yoneyama, A.: Matrix product solution to the reflection equation associated with a coideal subalgebra of \(U_q(A^{(1)}_{n-1})\). Lett. Math. Phys. 109, 2049–2067 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  24. Letzter, G.: Symmetric pairs for quantized enveloping algebras. J. Algebra 220(2), 729–767 (1999)

    Article  MathSciNet  Google Scholar 

  25. Kulish, L., Nepomechie, R.I.: Fusion procedure for open chains. J. Phys. A 25(9), 2533–2543 (1992)

    Article  MathSciNet  Google Scholar 

  26. Okado, M., Schilling, A.: Existence of Kirillov–Reshetikhin crystals for nonexceptional types. Represent. Theory 12, 186–207 (2008)

    Article  MathSciNet  Google Scholar 

  27. Regelskis, V., Vlaar, B.: Reflection matrices, coideal subalgebras and generalized Satake diagrams of affine type. arXiv:1602.08471

  28. Regelskis, V., Vlaar, B.: Solutions of the \(U_q(\mathfrak{\widehat{sl}} _N)\) reflection equations. J. Phys. A 51(34), 345204 (2018)

    Article  MathSciNet  Google Scholar 

  29. Sklyanin, E.: Boundary conditions for integrable quantum systems. J. Phys. A 21(10), 2375–2389 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  30. Smoktunowicz, A., Vendramin, L., Weston, R.: Combinatorial solutions to the reflection equation. J. Algebra 549, 268–290 (2020)

    Article  MathSciNet  Google Scholar 

  31. Watanabe, H.: A new tableau model for irreducible polynomial representations of the orthogonal group. J. Algebraic Combin. 58(1), 183–230 (2023)

    Article  MathSciNet  Google Scholar 

  32. Watanabe, H.: Crystal bases of modified \(\imath \)quantum groups of certain quasi-split types. Algebras Represent. Theor. (2023). https://doi.org/10.1007/s10468-023-10207-z

Download references

Acknowledgements

The authors would like to thank the referees for their careful readings and insightful comments. The authors also thank Atsuo Kuniba and Yasuhiko Yamada for interest to our work. M.O. is supported by JSPS KAKENHI Grant Number JP19K03426, and H.W. by JP21J00013. This work was partly supported by Osaka Central Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849).

Author information

Authors and Affiliations

  1. Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Hiroto Kusano

  2. Osaka Central Advanced Mathematical Institute and Department of Mathematics, Osaka Metropolitan University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Masato Okado & Hideya Watanabe

Authors
  1. Hiroto Kusano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masato Okado
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hideya Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hideya Watanabe.

Ethics declarations

Conflict of interest

The authors have no conflicts of interest to declare.

Additional information

Communicated by E. Smith.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kusano, H., Okado, M. & Watanabe, H. Kirillov–Reshetikhin Modules and Quantum K-matrices. Commun. Math. Phys. 405, 88 (2024). https://doi.org/10.1007/s00220-024-04975-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00220-024-04975-y

Search

Navigation