Skip to main content
SpringerLink
Log in

Cyclicity and R-matrices

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

Let \(S_1, \ldots , S_N\) simple finite-dimensional modules of a quantum affine algebra. We prove that if \(S_i\otimes S_j\) is cyclic for any \(i < j\) (i.e. generated by the tensor product of the highest weight vectors), then \(S_1\otimes \cdots \otimes S_N\) is cyclic. The proof is based on the study of R-matrices.

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

Similar content being viewed by others

References

  1. Beck, J.: Braid group action and quantum affine algebras. Commun. Math. Phys. 165(3), 555–568 (1994)

    Article  MathSciNet  Google Scholar 

  2. Beck, J., Chari, V., Pressley, A.: An algebraic characterization of the affine canonical basis. Duke Math. J. 99(3), 455–487 (1999)

    Article  MathSciNet  Google Scholar 

  3. Beck, J., Nakajima, H.: Crystal bases and two-sided cells of quantum affine algebras. Duke Math. J. 123(2), 335–402 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Chari, V.: Braid group actions and tensor products. Int. Math. Res. Not. 7, 357–382 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Chari, V., Pressley, A.: Quantum affine algebras. Commun. Math. Phys. 142(2), 261–283 (1991)

    Article  MathSciNet  Google Scholar 

  6. Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  7. Chari, V., Pressley, A.: Twisted quantum affine algebras. Commun. Math. Phys. 196(2), 461–476 (1998)

    Article  MathSciNet  Google Scholar 

  8. Costello, K., Witten, E., Yamazaki, M.: Gauge Theory and Integrability, I. arXiv:1709.09993

  9. Damiani, I.: La \({\cal{R}}\)-matrice pour les algèbres quantiques de type affine non tordu. Ann. Sci. Ecole Norm. Sup. (4) 31(4), 493–523 (1998)

    Article  MathSciNet  Google Scholar 

  10. Damiani, I.: From the Drinfeld realization to the Drinfeld–Jimbo presentation of affine quantum algebras: injectivity. Publ. Res. Inst. Math. Sci. 51, 131–171 (2015)

    Article  MathSciNet  Google Scholar 

  11. Frenkel, E., Mukhin, E.: Combinatorics of \(q\)-characters of finite-dimensional representations of quantum affine algebras. Commun. Math. Phys. 216(1), 23–57 (2001)

    Article  MathSciNet  Google Scholar 

  12. Frenkel, E., Reshetikhin, N.: The \(q\)-characters of representations of quantum affine algebras and deformations of \(W\)-algebras, recent developments in quantum affine algebras and related topics. Contemp. Math. 248, 163–205 (1999)

    Article  Google Scholar 

  13. Gurevich, M.: On restriction of unitarizable representations of general linear groups and the non-generic local Gan–Gross–Prasad conjecture. arXiv:1808.02640

  14. Hernandez, D.: The Kirillov–Reshetikhin conjecture and solutions of T-systems. J. Reine Angew. Math. 596, 63–87 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Hernandez, D.: Kirillov–Reshetikhin conjecture: the general case. Int. Math. Res. Not. 1, 149–193 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Hernandez, D.: Simple tensor product. Invent. Math. 181(3), 649–675 (2010)

    Article  MathSciNet  Google Scholar 

  17. Hernandez, D.: Avancées concernant les R-matrices et leurs applications (d’après Maulik-Okounkov, Kang- Kashiwara-Kim-Oh...), Sém. Bourbaki 69 ème année, 2016–2017, no. 1129, to appear in Astérisque

  18. Hernandez, D., Leclerc, B.: Cluster algebras and quantum affine algebras. Duke Math. J. 164(12), 2407–2460 (2015)

    Article  MathSciNet  Google Scholar 

  19. Hernandez, D., Leclerc, B.: A cluster algebra approach to q-characters of Kirillov–Reshetikhin modules. J. Eur. Math. Soc. 18(5), 1113–1159 (2016)

    Article  MathSciNet  Google Scholar 

  20. Jakelic, D., Moura, A.: Tensor products, characters, and blocks of finite-dimensional representations of quantum affine algebras at roots of unity. Int. Math. Res. Not. 2011(18), 4147–4199. arXiv:0909.2198v1

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

    Book  Google Scholar 

  22. Kashiwara, M.: On level-zero representations of quantized affine algebras. Duke Math. J. 112(1), 117–175 (2002)

    Article  MathSciNet  Google Scholar 

  23. Kang, S.-J., Kashiwara, M., Kim, M., Oh, S.-J.: Monoidal categorification of cluster algebras. J. Am. Math. Soc. 31, 349–426 (2018)

    Article  MathSciNet  Google Scholar 

  24. Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology. arXiv:1211.1287, to appear in Astérisque

  25. Nakajima, H.: t-analogs of q-characters of Kirillov–Reshetikhin modules of quantum affine algebras. Represent. Theory 7, 259–274 (2003)

    Article  MathSciNet  Google Scholar 

  26. Varagnolo, M., Vasserot, E.: Standard modules of quantum affine algebras. Duke Math. J. 111(3), 509–533 (2002)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank B. Leclerc and A. Moura for interesting comments and discussions. The author is supported by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement Number 647353 Qaffine.

Author information

Authors and Affiliations

  1. Sorbonne Paris Cité, Univ Paris Diderot, CNRS Institut de Mathématiques de Jussieu-Paris Rive Gauche UMR 7586, Batiment Sophie Germain, Case 7012, 75205, Paris Cedex 13, France

    David Hernandez

Authors
  1. David Hernandez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Hernandez.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hernandez, D. Cyclicity and R-matrices. Sel. Math. New Ser. 25, 19 (2019). https://doi.org/10.1007/s00029-019-0465-z

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s00029-019-0465-z

Keywords

Mathematics Subject Classification

Search

Navigation