Inventiones mathematicae

, Volume 211, Issue 2, pp 591–685 | Cite as

Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras

  • Seok-Jin Kang
  • Masaki Kashiwara
  • Myungho KimEmail author


Let J be a set of pairs consisting of good \(U'_q(\mathfrak g)\)-modules and invertible elements in the base field \(\mathbb C(q)\). The distribution of poles of normalized R-matrices yields Khovanov–Lauda–Rouquier algebras \(R^J(\beta )\) for each \(\beta \in \mathsf {Q}^+\). We define a functor \(\mathcal F_\beta \) from the category of graded \(R^J(\beta )\)-modules to the category of \(U'_q(\mathfrak g)\)-modules. The functor \(\mathcal F= \bigoplus _{\beta \in \mathsf {Q}^+} \mathcal {F}_\beta \) sends convolution products of finite-dimensional graded \(R^J(\beta )\)-modules to tensor products of finite-dimensional \(U'_q(\mathfrak g)\)-modules. It is exact if \(R^J\) is of finite type ADE. If \(V(\varpi _1)\) is the fundamental representation of \(U_q'({\widehat{\mathfrak {sl}}_N})\) of weight \(\varpi _1\) and \(J=\left\{ \bigl (V(\varpi _1), q^{2i} \bigr ) \mid i \in \mathbb Z \right\} \), then \(R^J\) is the Khovanov–Lauda–Rouquier algebra of type \(A_{\infty }\). The corresponding functor \(\mathcal {F}\) sends a finite-dimensional graded \(R^J\)-module to a module in \(\mathcal {C}_J\), where \(\mathcal {C}_J\) is the category of finite-dimensional integrable \(U_q'({\widehat{\mathfrak {sl}}_N})\)-modules M such that every composition factor of M appears as a composition factor of a tensor product of modules of the form \(V(\varpi _1)_{q^{2s}}\) \((s \in {\mathbb {Z}})\). Focusing on this case, we obtain an abelian rigid graded tensor category \({\mathcal T}_J\) by localizing the category of finite-dimensional graded \(R^J\)-modules. The functor \(\mathcal {F}\) factors through \({\mathcal T}_J\). Moreover, the Grothendieck ring of the category \(\mathcal {C}_J\) is isomorphic to the Grothendieck ring of \({\mathcal T}_J\) at \(q=1\).

Mathematics Subject Classification

81R50 16G 16T25 17B37 



We would like to express our gratitude to Bernard Leclerc for his kind explanations of his works and many fruitful discussions. The first and the third author gratefully acknowledge the hospitality of RIMS (Kyoto) during their visit in 2011 and 2012.


  1. 1.
    Akasaka, T., Kashiwara, M.: Finite-dimensional representations of quantum affine algebras. Publ. RIMS. Kyoto Univ. 33, 839–867 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ariki, S.: On the decomposition numbers of the Hecke algebra of \(G(M,1, n)\). J. Math. Kyoto Univ. 36, 789–808 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bernstein, I.N., Zelevinsky, A.V.: Induced representations of reductive \(p\)-adic groups. I. Ann. Sci. École. Norm. Sup. 10(4), 441–472 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brundan, J., Kleshchev, A.: Blocks of cyclotomic Hecke algebras and Khovanov–Lauda algebras. Invent. Math. 178, 451–484 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chari, V., Pressley, A.: A Guide to Quantum Groups. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  6. 6.
    Chari, V., Pressley, A.: Quantum affine algebras and affine Hecke algebras. Pac. J. Math. 174(2), 295–326 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cherednik, I.V.: A new interpretation of Gelfand–Tzetlin bases. Duke Math. J. 54, 563–577 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Date, E., Okado, M.: Calculation of excitation spectra of the spin model related with the vector representation of the quantized affine algebra of type \(A^{(1)}_n\). Int. J. Mod. Phys. A 9(3), 399–417 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ginzburg, V., Reshetikhin, N., Vasserot, E.: Quantum groups and flag varieties. A.M.S. Contemp. Math. 175, 101–130 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hernandez, D.: Algebraic approach to \(q, t\)-characters. Adv. Math. 187, 1–52 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hernandez, D., Leclerc, B.: Cluster algebras and quantum affine algebras. Duke Math. J. 154(2), 265–341 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hernandez, D., Leclerc, B.: Quantum Grothendieck rings and derived Hall algebras. J. Reine Angew. Math. doi: 10.1515/crelle-2013-0020
  13. 13.
    Jimbo, M.: A \(q\)-analogue of \(U(\mathfrak{gl}_{N+1})\), Hecke algebra, and the Yang–Baxter equation. Lett. Math. Phys. 11, 247–252 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kac, V.: Infinite Dimensional Lie algebras, 3rd edn. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kang, S.-J., Kashiwara, M.: Categorification of highest weight modules via Khovanov–Lauda–Rouquier algebras. Invent. Math. 190, 699–742 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kang, S.-J., Park, E.: Irreducible modules over Khovanov–Lauda–Rouquier algebras of type \(A_n\) and semistandard tableaux. J. Algebra 339, 223–251 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kashiwara, M.: Crystalizing the \(q\)-analogue of universal enveloping algebras. Commun. Math. Phys. 133, 249–260 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kashiwara, M.: On crystal bases of the \(q\)-analogue of universal enveloping algebras. Duke Math. J. 63, 465–516 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kashiwara, M.: Crystal bases of modified quantized enveloping algebra. Duke Math. J. 73, 383–413 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kashiwara, M.: On level zero representations of quantum affine algebras. Duke Math. J. 112, 117–175 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kashiwara, M., Schapira, P.: Categories and Sheaves, Grundlehren der mathematischen Wissenschaften 332. Springer, Berlin (2006)Google Scholar
  22. 22.
    Kato, S.: Poincaré–Birkhoff–Witt bases and Khovanov–Lauda–Rouquier algebras. Duke Math. J. 163(3), 619–663 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups I. Represent. Theory 13, 309–347 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups II. Trans. Am. Math. Soc. 363, 2685–2700 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Kim, M.: Khovanov–Lauda–Rouquier Algebras and R-Matrices, Ph.D. thesis, Seoul National University (2012)Google Scholar
  26. 26.
    Kleshchev, A.S., Mathas, A., Ram, A.: Universal graded Specht modules for cyclotomic Hecke algebras. Proc. Lond. Math. Soc. (3) 105(6), 1245–1289 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lascoux, A., Leclerc, B., Thibon, J.-Y.: Hecke algebras at roots of unity and crystal bases of quantum affine algebras. Commun. Math. Phys. 181, 205–263 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Lauda, A., Vazirani, M.: Crystals from categorified quantum groups. Adv. Math. 228, 803–861 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Lusztig, G.: Introduction to Quantum Groups. Birkhöser, Boston (1993)zbMATHGoogle Scholar
  30. 30.
    McNamara, P.: Finite dimensional representations of Khovanov–Lauda–Rouquier algebras I: finite type. J. Reine Angew. Math. 707, 103–124 (2015)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Nakajima, H.: Quiver varieties and \(t\)-analogue of \(q\)-characters of quantum affine algebras. Ann. Math. 160, 1057–1097 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Popescu, N.: Abelian Categories with Applications to Rings and Modules, L.M.S. Monographs, vol. 3, London Mathematical Society (1973)Google Scholar
  33. 33.
    Rouquier, R.: 2-Kac-Moody algebras, arXiv:0812.5023v1
  34. 34.
    Rouquier, R.: Quiver Hecke algebras and 2-Lie algebras. Algebra Colloq. 19, 359–410 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Varagnolo, M., Vasserot, E.: Perverse sheaves and quantum Grothendieck rings. In: Studies in Memory of Issai Schur, Prog. Math., vol. 210. Birkhäuser, pp. 345–365 (2002)Google Scholar
  36. 36.
    Varagnolo, M., Vasserot, E.: Canonical bases and KLR algebras. J. Reine Angew. Math. 659, 67–100 (2011)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Vazirani, M.: Parameterizing Hecke algebra modules: Bernstein–Zelevinsky multisegments, Kleshchev multipartitions, and crystal graphs. Tranform. Groups 7(3), 267–303 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zelevinsky, A.V.: Induced representations of reductive \(p\)-adic groups. II. On irreducible representations of \(GL(n)\). Ann. Sci. École. Norm. Sup. 13(2), 165–210 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Joeun Mathematical Research InstituteSeoulKorea
  2. 2.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  3. 3.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulKorea
  4. 4.Department of MathematicsKyung Hee UniversitySeoulKorea

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