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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 Springer Proceedings in Mathematics & Statistics book series (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).

Keywords

  • Cluster Algebras
  • Monoidal Categorification
  • Dynkin Quiver
  • Tensor Category
  • Dominant Monomer

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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