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Graded quiver varieties and singularities of normalized R-matrices for fundamental modules

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Abstract

We present a simple unified formula expressing the denominators of the normalized R-matrices between the fundamental modules over the quantum loop algebras of type \({\mathsf {ADE} }\). It has an interpretation in terms of representations of Dynkin quivers and can be proved in a unified way using geometry of the graded quiver varieties. As a by-product, we obtain a geometric interpretation of Kang–Kashiwara–Kim’s generalized quantum affine Schur–Weyl duality functor when it arises from a family of the fundamental modules. We also study several cases when the graded quiver varieties are isomorphic to unions of the graded nilpotent orbits of type \(\mathsf {A} \).

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Notes

  1. In [24], the affinization is defined in terms of the Chevalley type generators of the algebra \(U'_q({\widehat{{\mathfrak {g}}}})\). One can easily see that it coincides with our affinization in Definition 2.4 under the isomorphism \(U_{q}(L{\mathfrak {g}}) \cong U_{q}^{\prime }({\widehat{{\mathfrak {g}}}}) / \langle q^{c} -1 \rangle \) in [3].

References

  1. Akasaka, T., Kashiwara, M.: Finite-dimensional representations of quantum affine algebras. Publ. Res. Inst. Math. Sci. 33(5), 839–867 (1997)

    Article  MathSciNet  Google Scholar 

  2. Assem, I., Simson, D., Skowroński, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  3. Beck, J.: Braid group action and quantum affine algebras. Commun. Math. Phys. 165(3), 555–568 (1994)

    Article  MathSciNet  Google Scholar 

  4. Chari, V.: Braid group actions and tensor products. Int. Math. Res. Not. 2002(7), 357–382 (2002)

    Article  MathSciNet  Google Scholar 

  5. Chari, V., Pressley, A.: Quantum affine algebras and their representations. In: Representations of Groups (Banff, AB, 1994), CMS Conf. Proc., vol. 16, pp. 59–78. Amer. Math. Soc. Providence (1995)

  6. Chari, V., Pressley, A.: Weyl modules for classical and quantum affine algebras. Represent. Theory 5, 191–223 (2001)

    Article  MathSciNet  Google Scholar 

  7. Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birkhauser Boston, Inc., Boston (1997)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  9. Frenkel, E., Mukhin, E.: Combinatorics of \(q\)-characters of finite-dimensional representations of quantum affine algebras. Commun. Math. Phys. 216(1), 23–57 (2001)

    Article  MathSciNet  Google Scholar 

  10. Frenkel, E., Reshetikhin, N.: The \(q\)-characters of representations of quantum affine algebras and deformations of \({\cal{W}}\)-algebras. In: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Contemp. Math., vol. 248, pp. 163–205. Amer. Math. Soc. Providence (1999)

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

  12. Fujita, R.: Geometric realization of Dynkin quiver type quantum affine Schur–Weyl duality. Int. Math. Res. Not. (2018). https://doi.org/10.1093/imrn/rny226

    Article  MATH  Google Scholar 

  13. Gabriel, P.: Unzerlegbare Darstellungen. I. Manuscr. Math. 6, 71–103 (1972); correction, ibid. 6, 309 (1972)

  14. Gabriel, P.: Auslander-Reiten sequences and representation-finite algebras. In: Representation Theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 831, pp. 1–71. Springer, Berlin (1980)

  15. Ginzburg, V., Reshetikhin, N., Vasserot, E.: Quantum groups and flag varieties. In: Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups (South Hadley, MA, 1992), Contemp. Math., no. 175, pp. 101–130. Amer. Math. Soc. Providence (1994)

  16. Happel, D.: On the derived category of a finite-dimensional algebra. Comment. Math. Helv. 62(3), 339–389 (1987)

    Article  MathSciNet  Google Scholar 

  17. Happel, D.: Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Mathematical Society Lecture Note Series, vol. 119. Cambridge University Press, Cambridge (1988)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Kang, S.-J., Kashiwara, M., Kim, M.: Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, II. Duke Math. J. 164(8), 1549–1602 (2015)

    Article  MathSciNet  Google Scholar 

  20. Kang, S.-J., Kashiwara, M., Kim, M.: Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras. Invent. Math. 211(2), 591–685 (2018)

    Article  MathSciNet  Google Scholar 

  21. Kang, S.-J., Kashiwara, M., Kim, M., Oh, S.-J.: Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, III. Proc. Lond. Math. Soc. (3) 111(2), 420–444 (2015)

    Article  MathSciNet  Google Scholar 

  22. Kang, S.-J., Kashiwara, M., Kim, M., Oh, S.-J.: Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, IV. Sel. Math. (N.S.) 22(4), 1987–2015 (2016)

    Article  MathSciNet  Google Scholar 

  23. Kashiwara, M.: Crystal bases of modified quantized enveloping algebra. Duke Math. J. 73(2), 383–413 (1994)

    Article  MathSciNet  Google Scholar 

  24. Kashiwara, M.: On level-zero representations of quantized affine algebras. Duke Math. J. 112(1), 117–175 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Kashiwara, M., Kim, M., Oh, S.-j., Park, E.: Cluster algebra structures on module categories over quantum affine algebras. Preprint. arXiv:1904.01264

  27. Kashiwara, M., Oh, S.-J.: Categorical relations between Langlands dual quantum affine algebras: doubly laced types. J. Algebr. Comb. 49(4), 401–435 (2019)

    Article  MathSciNet  Google Scholar 

  28. Kato, S.: Poincare–Birkhoff–Witt bases and Khovanov–Lauda–Rouquier algebras. Duke Math. J. 163(3), 619–663 (2014)

    Article  MathSciNet  Google Scholar 

  29. Keller, B., Scherotzke, S.: Graded quiver varieties and derived categories. J. Reine Angew. Math. 713, 85–127 (2016)

    MathSciNet  MATH  Google Scholar 

  30. Khovanov, M., Lauda, A.: A diagrammatic approach to categorification of quantum groups. I. Represent. Theory 13, 309–347 (2009)

    Article  MathSciNet  Google Scholar 

  31. Leclerc, B., Plamondon, P.-G.: Nakajima varieties and repetitive algebras. Publ. Res. Inst. Math. Sci. 49(3), 531–561 (2013)

    Article  MathSciNet  Google Scholar 

  32. Nakajima, H.: Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Am. Math. Soc. 14(1), 145–238 (2001)

    Article  MathSciNet  Google Scholar 

  33. Nakajima, H.: Quiver varieties and tensor products. Invent. Math. 146(2), 399–449 (2001)

    Article  MathSciNet  Google Scholar 

  34. Nakajima, H.: Extremal weight modules of quantum affine algebras. In: Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., vol. 40, pp. 343–369. Math. Soc. Japan, Tokyo (2004)

  35. Oh, S.-J.: The denominators of normalized \(R\)-matrices of types \(A_{2n-1}^{(2)}\), \(A_{2n}^{(2)}\), \(B_n^{(1)}\) and \(D_{n+1}^{(2)}\). Publ. Res. Inst. Math. Sci. 51(4), 709–744 (2015)

    Article  MathSciNet  Google Scholar 

  36. Oh, S.-J., Scrimshaw, T.: Categorical relations between Langlands dual quantum affine algebras: exceptional cases. Commun. Math. Phys. 368(1), 295–367 (2019)

    Article  MathSciNet  Google Scholar 

  37. Oh, S.-J., Scrimshaw, T.: Correction to: Categorical relations between Langlands dual quantum affine algebras: exceptional cases. Commun. Math. Phys. 371(2), 833–837 (2019)

    Article  MathSciNet  Google Scholar 

  38. Rouquier, R.: 2-Kac–Moody algebras. Preprint. arXiv:0812.5023

  39. Varagnolo, M., Vasserot, E.: Standard modules of quantum affine algebras. Duke Math. J. 111(3), 509–533 (2002)

    Article  MathSciNet  Google Scholar 

  40. Varagnolo, M., Vasserot, E.: Canonical bases and KLR-algebras. J. Reine Angew. Math. 659, 67–100 (2011)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The author is grateful to Se-jin Oh for his interest in this paper and for answering the author’s questions on his papers. The author was supported by Grant-in-Aid for JSPS Research Fellow (No. 18J10669), and by JSPS Overseas Research Fellowships during the revision.

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  1. Research Institute for Mathematical Sciences, Kyoto University, Oiwake-Kitashirakawa, Sakyo, Kyoto, 606-8502, Japan

    Ryo Fujita

  2. Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris, 75013, Paris, France

    Ryo Fujita

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Fujita, R. Graded quiver varieties and singularities of normalized R-matrices for fundamental modules. Sel. Math. New Ser. 28, 2 (2022). https://doi.org/10.1007/s00029-021-00715-5

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