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Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification pp 37–65Cite as

Birkhäuser

Quantum Affine Algebras and Cluster Algebras

Part of the Progress in Mathematics book series (PM,volume 337)

Abstract

This article is an extended version of the minicourse given by the second author at the summer school of the conference Interactions of quantum affine algebras with cluster algebras, current algebras and categorification, held in June 2018 in Washington. The aim of the minicourse, consisting of three lectures, was to present a number of results and conjectures on certain monoidal categories of finite-dimensional representations of quantum affine algebras, obtained by exploiting the fact that their Grothendieck rings have the natural structure of a cluster algebra.

To Vyjayanthi Chari on her birthday

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Notes

  1. 1.

    In these lectures, we will only consider type I representations of quantum enveloping algebras. All representations can be obtained from the type I representations by twisting with some signs, see e.g. [4, §10.1].

References

  1. M. Brito, V. Chari, Tensor products and q-characters of HL-modules and monoidal categorifications, J. Éc. polytech. math. 6 (2019), 581–619.

    CrossRef  MathSciNet  Google Scholar 

  2. V. Chari, Braid group actions and tensor products, Int. Math. Res. Not. (2002), no. 7, 357–382.

    Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

  4. V. Chari, A. Pressley, A guide to quantum groups. Cambridge University Press 1994.

    Google Scholar 

  5. V. Chari, A. Pressley, Minimal affinizations of representations of quantum groups: the simply laced case, J. Algebra 184 (1996), 1–30.

    CrossRef  MathSciNet  Google Scholar 

  6. V. Chari, A. Pressley, Quantum affine algebras and affine Hecke algebras, Pacific J. Math. 174 (1996), 295–326.

    CrossRef  MathSciNet  Google Scholar 

  7. V. Chari, A. Pressley, Factorizations of representations of quantum affine algebras in Modular Interfaces, (Riverside, Calif. 1995), AMS/IP Stud. Adv. Math. 4, Amer. Math. Soc., Providence, 1997, 33–40.

    Google Scholar 

  8. H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potential and their representations I: Mutations, Selecta Math., 14 (2008), 59–119.

    CrossRef  MathSciNet  Google Scholar 

  9. H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potential and their representations II: Applications to cluster algebras, J. Amer. Math. Soc. 23 (2010), 749–790.

    CrossRef  MathSciNet  Google Scholar 

  10. R. Fujita, Affine highest weight categories and quantum affine Schur-Weyl duality of Dynkin quiver types, arXiv:1710.11288

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

  13. E. Frenkel, N. Reshetikhin, The q-characters of representations of quantum affine algebras, Recent developments in quantum affine algebras and related topics, Contemp. Math. 248 (1999), 163–205.

    CrossRef  Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

  16. S. Fomin, A. Zelevinsky, Cluster algebras: notes for the CDM-03 conference, in Current developments in mathematics, 2003, 1–34, Int. Press, Somerville, MA, 2003.

    MATH  Google Scholar 

  17. C. Geiss, B. Leclerc, J. Schröer, Factorial cluster algebras, Doc. Math. 18 (2013), 249–274.

    MathSciNet  MATH  Google Scholar 

  18. V. Ginzburg, N. Reshetikhin, E. Vasserot, Quantum groups and flag varieties, in Mathematical aspects of conformal and topological field theories and quantum groups (South Hadley, MA, 1992), 101–130, Contemp. Math., 175, Amer. Math. Soc., Providence, RI, 1994.

    Google Scholar 

  19. M. Gekhtman, M. Shapiro, A. Vainshtein, Cluster algebras and Poisson geometry, AMS Math. Survey and Monographs 167, AMS 2010.

    Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

  23. D. Hernandez, B. Leclerc, Monoidal categorifications of cluster algebras of type A and D, in Symmetries, integrable systems and representations, (K. Iohara, S. Morier-Genoud, B. Rémy, eds.), Springer proceedings in mathematics and statistics 40, 2013, 175–193.

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

  26. D. Hernandez, B. Leclerc, Cluster algebras and category \(\mathcal {O}\) for representations of Borel subalgebras of quantum affine algebras, Algebra Number Theory 10 (2016), 2015–2052.

    CrossRef  MathSciNet  Google Scholar 

  27. D. Hernandez, H. Oya, Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm, Adv. Math. 374 (2019), 192–272.

    CrossRef  MathSciNet  Google Scholar 

  28. S.-J. Kang, M. Kashiwara, M. Kim, Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, II, Duke Math. J. 164 (2015), 1549–1602.

    CrossRef  MathSciNet  Google Scholar 

  29. S.-J. Kang, M. Kashiwara, M. Kim, S.-J. Oh, Simplicity of heads and socles of tensor products, Compos. Math. 151 (2015), 377–396.

    CrossRef  MathSciNet  Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

  31. M. Kashiwara, M. Kim, S.-J. Oh, Monoidal categories of modules over quantum affine algebras of type A and B, Proc. London Math. Soc. 118 (2019) 43–77.

    CrossRef  MathSciNet  Google Scholar 

  32. M. Kashiwara, S.-j. Oh, Categorical relations between Langlands dual quantum affine algebras: doubly laced types, to appear in J. Algebraic Combin. 49 (2019), 401–435. https://doi.org/10.1007/s10801-018-0829-z

    CrossRef  MathSciNet  Google Scholar 

  33. A. Kuniba, T. Nakanishi, J. Suzuki, Functional relations in solvable lattice models: I. Functional relations and representation theory, Int. J. Mod. Phys. A9 (1994), 5215–5266.

    CrossRef  MathSciNet  Google Scholar 

  34. B. Leclerc, Imaginary vectors in the dual canonical basis of U q(n), Transform. Groups 8 (2003), 95–104.

    CrossRef  MathSciNet  Google Scholar 

  35. B. Leclerc, Quantum loop algebras, quiver varieties, and cluster algebras, in Representations of Algebras and Related Topics, (A. Skowroński and K. Yamagata, eds.), European Math. Soc. Series of Congress Reports, 2011, 117–152.

    Google Scholar 

  36. E. Lapid, A. Minguez, Geometric conditions for \(\square \)-irreducibility of certain representations of the general linear group over a non-Archimedean local field, Adv. Math. 339 (2018), 113–190.

    CrossRef  MathSciNet  Google Scholar 

  37. G. Lusztig, On quiver varieties, Adv. Math. 136 (1998), 141–182.

    CrossRef  MathSciNet  Google Scholar 

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

    CrossRef  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  40. Fan Qin, Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. J. 166 (2017), 2337–2442.

    CrossRef  MathSciNet  Google Scholar 

  41. A. Savage, P. Tingley, Quiver Grassmannians, quiver varieties and preprojective algebras, Pacific J. Math. 251 (2011), 393–429.

    CrossRef  MathSciNet  Google Scholar 

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Acknowledgements

D. Hernandez is supported in part by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement number 647353 QAffine.

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Authors and Affiliations

  1. Université de Paris, Univ Paris Diderot, CNRS Institut de Mathématiques de Jussieu-Paris Rive Gauche UMR 7586, Paris, France

    David Hernandez

  2. Institut Universitaire de France, Paris, France

    David Hernandez

  3. Normandie Univ, UNICAEN, CNRS, LMNO, Caen, France

    Bernard Leclerc

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  1. David Hernandez
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Correspondence to Bernard Leclerc .

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Editors and Affiliations

  1. Department of Mathematics, University of California, Riverside, Riverside, CA, USA

    Jacob Greenstein

  2. IMJ-PRG, Université de Paris, Paris, France

    Prof. David Hernandez

  3. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Kailash C. Misra

  4. Department of Mathematics, Catholic University of America, Washington D.C., WA, USA

    Prasad Senesi

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Hernandez, D., Leclerc, B. (2021). Quantum Affine Algebras and Cluster Algebras. In: Greenstein, J., Hernandez, D., Misra, K.C., Senesi, P. (eds) Interactions of Quantum Affine Algebras with Cluster Algebras, Current Algebras and Categorification. Progress in Mathematics, vol 337. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-63849-8_2

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