Skip to main content
SpringerLink
Log in

Toroidal Grothendieck rings and cluster algebras

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We study deformations of cluster algebras with several quantum parameters, called toroidal cluster algebras, which naturally appear in the study of Grothendieck rings of representations of quantum affine algebras. In this context, we construct toroidal Grothendieck rings and we establish these are flat deformations of Grothendieck rings. We prove that for a family of monoidal categories \({\mathscr {C}}_1\) of simply-laced quantum affine algebras categorifying finite-type cluster algebras, the toroidal Grothendieck ring has a natural structure of a toroidal cluster algebra.

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

Notes

  1. This is obtained from the quasi-commutation relation in (9), but with a sign change, so that it agrees with the formula in [26] corresponding to the case \( a = 0\).

  2. Here, we are using the same product as in [26], which differs from the one in [22] by replacing t with \( t^{-1}\). This change amounts to a sign change in the RHS of the formulas for \( {\mathcal {N}}_a(i,p; j,s)\).

  3. Note that in the example of Sect. 2.2 the parameters are actually denoted by \( t_1\) and \( t_2\) respectively, following the notation introduced in that section.

  4. Again, under a sign change. Here, with the purpose that the result is compatible with the quasi-commutation product between the variables \( Y_{i,p}^{\pm 1}\).

  5. Note that in [30] the authors consider the same quantum torus up to a minus sign. Moreover, the quantum Cartan matrices coincides up to a swap of rows (resp. columns) 1 and 2, and its inverse \({\widetilde{C}}(z)\) is expanded for negative z.

References

  1. Bittmann, L.: A quantum cluster algebra approach to representations of simply-laced quantum affine algebras. Math. Z (2020). arXiv:1911.13110

  2. Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras III: Upper bounds and double Bruhat cells. Duke Math. J. 126(1), 1–52 (2005)

    Article  MathSciNet  Google Scholar 

  3. Berenstein, A., Zelevinsky, A.: Quantum cluster algebras. Adv. Math. 195(2), 405–455 (2005)

    Article  MathSciNet  Google Scholar 

  4. Brito, M., Chari, V.: Tensor products and q-characters of HL-modules and monoidal categorifications. J. Éc. Polytech. Math. 6, 581–619 (2019)

    Article  MathSciNet  Google Scholar 

  5. Chari, V., Hernandez, D.: Beyond Kirillov-Reshetikhin modules. In: Quantum Affine Algebras, Extended Affine Lie Algebras, and Their Applications. Contemp. Math. vol. 506, , pp. 49–81. Amer. Math. Soc., Providence, RI (2010)

  6. Chari, V., Moura, A.A.: Characters and blocks for finite-dimensional representations of quantum affine algebras. Int. Math. Res. Not. 2005(5), 257–298 (2005)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  8. Cautis, S., Williams, H.: Cluster theory of the coherent Satake category. J. Am. Math. Soc. 32, 709–778 (2019)

    Article  MathSciNet  Google Scholar 

  9. Davison, B.: Positivity for quantum cluster algebras. Ann. Math. (2) 187(1), 157–219 (2018)

    Article  MathSciNet  Google Scholar 

  10. Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002)

    Article  MathSciNet  Google Scholar 

  11. Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154(1), 63–121 (2003)

    Article  MathSciNet  Google Scholar 

  12. Frenkel, E., Reshetikhin, N.: Deformations of \(W\)-Algebras associated to simple Lie algebras. Commun. Math. Phys. 197(1), 1–32 (1998)

    Article  MathSciNet  Google Scholar 

  13. Frenkel, E., Reshetikhin, N.: The \(q\)-characters of representations of quantum affine algebras and deformations of \(W\)-algebras. In: Recent Developments in Quantum Affine Algebras and related topics (Raleigh, N.C.), 1998, Contemp. Math. vol. 248, pp. 163–205 (1999)

  14. Fujita, R.: Affine highest weight categories and quantum affine Schur–Weyl duality of Dynkin quiver types (2020). arXiv:1710.11288

  15. Fujita, R.: Geometric realization of Dynkin quiver type quantum affine Schur-Weyl duality. Int. Math. Res. Not. (22), 8353–8386 (2020)

  16. Gautam, S., Laredo, V.T.: Meromorphic tensor equivalence for Yangians and quantum loop algebras. Publ. Math. Inst. Hautes Études Sci. 125, 267–337 (2017)

    Article  MathSciNet  Google Scholar 

  17. Geiss, C., Leclerc, B., Schroer, J.: Cluster structures on quantum coordinate rings. Sel. Math. 19(2), 337–397 (2013)

    Article  MathSciNet  Google Scholar 

  18. Geiss, C., Leclerc, B., Schroer, J.: Quantum cluster algebras and their specializations. J. Algebra 558, 411–422 (2020)

    Article  MathSciNet  Google Scholar 

  19. Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster Algebras and Poisson geometry, Mathematical Surveys and Monographs 167, 246 pp. American Mathematical Society, Providence, RI (2010)

    Book  Google Scholar 

  20. Goodearl, K.R., Yakimov, M.T.: Quantum cluster algebra structures on quantum nilpotent algebras. Mem. Am. Math. Soc. 247(1169), vii+119 pp (2017)

    MathSciNet  MATH  Google Scholar 

  21. Goodearl, K.R., Yakimov, M.T.: Cluster algebra structures on Poisson nilpotent algebras (2020). arXiv:1801.01963

  22. Hernandez, D.: Algebraic approach to \(q, t\)-characters. Adv. Math. 187, 1–52 (2004)

    Article  MathSciNet  Google Scholar 

  23. Hernandez, D.: Monomials of \(q\) and \(q, t\)-characters for non simply-laced quantum affinizations. Math. Z. 250(2), 443–473 (2005)

    Article  MathSciNet  Google Scholar 

  24. Hernandez, D.: Avancées concernant les R-matrices et leurs applications (d’après Maulik-Okounkov, Kang-Kashiwara-Kim-Oh...). Sém. Bourbaki 1129 Astérisque 407, 297–331 (2019)

    Google Scholar 

  25. Hernandez, D., Leclerc, B.: Cluster algebras and quantum affine algebras. Duke Math. J. 154(2), 265–341 (2010)

    Article  MathSciNet  Google Scholar 

  26. Hernandez, D., Leclerc, B.: Quantum Grothendieck rings and derived Hall algebras. J. Reine Angew. Math. 701, 77–126 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Hernandez, D., Leclerc, B.: Monoidal categorifications of cluster algebras of type A and D. In: Symmetries, integrable systems and representations, Springer Proc. Math. Stat., vol. 40, pp. 175–193 (2013)

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

    Article  MathSciNet  Google Scholar 

  29. Hernandez, D., Leclerc, B.: Quantum affine algebras and cluster algebras. In: Progress in Mathematics, Honor of V, Chari, vol. 337 (2021)

  30. Hernandez, D., Oya, H.: Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan–Lusztig algorithm. Adv. Math. 347, 192–272 (2019)

    Article  MathSciNet  Google Scholar 

  31. Hu, N., Pei, Y.: Notes on two-parameter groups (I). Sci. China Ser. A 51(6), 1101–1110 (2008)

    Article  MathSciNet  Google Scholar 

  32. Hu, N., Pei, Y., Rosso, M.: Multi-parameter quantum groups and quantum shuffles. In: Quantum Affine Algebras, Extended Affine Lie Algebras, and Their Applications. Contemp. Math., vol. 506, pp. 145–171 (2010)

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

    Book  Google Scholar 

  34. Kang, S.-J., Kashiwara, M., Kim, M., Oh, S.-J.: Monoidal categorification of cluster algebras. J. Am. Math. Soc. 31(2), 349–426 (2018)

    Article  MathSciNet  Google Scholar 

  35. Kashiwara, M.: Crystal Bases and Categorifications, Proceedings of the ICM 2018, vol. I, pp. 249–258 (2018)

  36. Kuniba, A., Nakanishi, T., Suzuki, J.: T-systems and Y-systems in integrable systems. J. Phys. A 44(10), 146 (2011)

    Article  MathSciNet  Google Scholar 

  37. Lee, K., Schiffler, R.: Positivity for cluster algebras. Ann. Math. (2) 182(1), 73–125 (2015)

    Article  MathSciNet  Google Scholar 

  38. Nakajima, H.: Quiver varieties and \(t\)-analogs of \(q\)-characters of quantum affine algebras. Ann. Math. (2) 160(3), 1057–1097 (2004)

    Article  MathSciNet  Google Scholar 

  39. 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 

  40. Nakajima, H.: Quiver varieties and cluster algebras. Kyoto J. Math. 51, 71–126 (2011)

    MathSciNet  MATH  Google Scholar 

  41. Oh, S.-J., Suh, U.: Twisted and folded Auslander–Reiten quiver and applications to the representation theory of quantum affine algebras. J. Algebra 535, 53–132 (2019)

    Article  MathSciNet  Google Scholar 

  42. Okounkov, A.: On the Crossroads of Enumerative Geometry and Geometric Representation Theory. Proceedings of the ICM 2018, vol. I, pp. 839–867 (2018)

  43. Qin, F.: Triangular bases in quantum cluster algebras and monoidal categorification conjectures. Duke Math. J. 166(12), 2337–2442 (2017)

    Article  MathSciNet  Google Scholar 

  44. Varagnolo, M., Vasserot, E.: Perverse sheaves and quantum Grothendieck rings. Studies in Memory of Issai Schur (Chevaleret/Rehovot, 2000), Progr. Math., vol. 210, pp. 345–365. Birkhäuser, Boston (2003)

Download references

Acknowledgements

The authors would like to thank the referee for his comments and remarks. The authors would like to thank also Martina Lanini, Bernard Leclerc and Bernhard Keller for discussions and references. The authors were 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. Université de Paris and Sorbonne Université, CNRS, IMJ-PRG, F-75006, Paris, France

    Laura Fedele & David Hernandez

Authors
  1. Laura Fedele
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Hernandez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laura Fedele.

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

Fedele, L., Hernandez, D. Toroidal Grothendieck rings and cluster algebras. Math. Z. 300, 377–420 (2022). https://doi.org/10.1007/s00209-021-02780-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-021-02780-0

Search

Navigation