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Symmetries, Integrable Systems and Representations pp 175–193Cite as

Monoidal Categorifications of Cluster Algebras of Type A and D

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 40))

Abstract

In this note, we introduce monoidal subcategories of the tensor category of finite-dimensional representations of a simply-laced quantum affine algebra, parametrized by arbitrary Dynkin quivers. For linearly oriented quivers of types A and D, we show that these categories provide monoidal categorifications of cluster algebras of the same type. The proof is purely representation-theoretical, in the spirit of Hernandez and Leclerc (Duke Math. J. 154, 265–341, 2010).

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References

  1. Cerulli Irelli, G., Keller, B., Labardini-Fragoso, D., Plamondon, P.: Linear independence of cluster monomials for skew-symmetric cluster algebras. arXiv:1203.1307

  2. 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 (2010)

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Fomin, S., Zelevinsky, A.: Cluster algebras: notes for the CDM-03 conference. In: Current Developments in Mathematics, pp. 1–34. Int. Press, Somerville (2003)

    Google Scholar 

  5. Fomin, S., Zelevinsky, A.: Cluster algebras IV: coefficients. Compos. Math. 143, 112–164 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fomin, S., Shapito, M., Thurston, D.: Cluster algebras and triangulated surfaces. I. Cluster complexes. Acta Math. 201(1), 83–146 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Frenkel, E., Reshetikhin, N.: The q-characters of representations of quantum affine algebras. In: Recent Developments in Quantum Affine Algebras and Related Topics. Contemp. Math., vol. 248, pp. 163–205 (1999)

    Chapter  Google Scholar 

  9. Geiss, C., Leclerc, B., Schröer, J.: Kac-Moody groups and cluster algebras. Adv. Math. 228, 329–433 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Geiss, C., Leclerc, B., Schröer, J.: Factorial cluster algebras. arXiv:1110.1199

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Hernandez, D.: On minimal affinizations of representations of quantum groups. Commun. Math. Phys. 277(1), 221–259 (2007)

    Article  Google Scholar 

  13. Hernandez, D.: Smallness problem for quantum affine algebras and quiver varieties. Ann. Sci. Éc. Norm. Super. 41(2), 271–306 (2008)

    MATH  Google Scholar 

  14. Hernandez, D.: Simple tensor products. Invent. Math. 181, 649–675 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Hernandez, D., Leclerc, B.: Quantum Grothendieck rings and derived Hall algebras. arXiv:1109.0862

  17. Keller, B.: Algèbres amassées et applications (d’après Fomin-Zelevinsky, …). In: Séminaire Bourbaki. Vol. 2009/2010. Exposés 1012-1026. Astérisque, vol. 339, pp. 63–90 (2011). Exp. No. 1014, vii

    Google Scholar 

  18. Leclerc, B.: Quantum loop algebras, quiver varieties, and cluster algebras. In: Skowroński, A., Yamagata, K. (eds.) Representations of Algebras and Related Topics. EMS Series of Congress Reports, pp. 117–152 (2011)

    Chapter  Google Scholar 

  19. Mukhin, E., Young, C.A.S.: Extended T-systems. Sel. Math. New Ser. 18, 591–631 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Qin, F.: Algèbres amassées quantiques acycliques. PhD thesis, Université Paris 7, May (2012)

    Google Scholar 

  22. Yang, S.-W.: Combinatorial expressions for F-polynomials in classical types. J. Comb. Theory, Ser. A 119(3), 747–764 (2012)

    Article  MATH  Google Scholar 

  23. Yang, S.-W., Zelevinsky, A.: Cluster algebras of finite type via Coxeter elements and principal minors. Transform. Groups 13(3–4), 855–895 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The first author would like to thank A. Zelevinsky for explaining the results in [22, 23]. The authors are grateful to the referee for useful comments.

Author information

Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, CNRS UMR 7586, Université Paris 7, 175 Rue du Chevaleret, 75013, Paris, France

    David Hernandez

  2. UMR 6139 LMNO, Université de Caen Basse-Normandie, 14032, Caen, France

    Bernard Leclerc

  3. CNRS UMR 6139 LMNO, Institut Universitaire de France, 14032, Caen, France

    Bernard Leclerc

Authors
  1. David Hernandez
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  2. Bernard Leclerc
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Corresponding author

Correspondence to David Hernandez .

Editor information

Editors and Affiliations

  1. Inst. Camille Jordan, UMR5028 CNRS, Université Claude Bernard Lyon 1, Villeurbanne Cedex, 69622, France

    Kenji Iohara

  2. Inst. de Mathém. de Jus., UMR 7586 CNRS, Université Pierre et Marie Curie, Place Jussieu, Case 247 4, Paris, 75252, France

    Sophie Morier-Genoud

  3. Inst. Camille Jordan, UMR5028 CNRS, Université Claude Bernard Lyon I, Villeurbanne Cedex, 69622, France

    Bertrand Rémy

Additional information

To M. Jimbo on his 60th birthday.

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Hernandez, D., Leclerc, B. (2013). Monoidal Categorifications of Cluster Algebras of Type A and D . In: Iohara, K., Morier-Genoud, S., Rémy, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/978-1-4471-4863-0_8

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