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A categorical action on quantized quiver varieties

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Abstract

In this paper, we describe a categorical action of any symmetric Kac–Moody algebra on a category of quantized coherent sheaves on Nakajima quiver varieties. By “quantized coherent sheaves,” we mean a category of sheaves of modules over a deformation quantization of the natural symplectic structure on quiver varieties. This action is a direct categorification of the geometric construction of universal enveloping algebras by Nakajima.

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References

  1. Beilinson, A., Bernstein, J.: Localisation de \(\mathfrak{g}\)-modules. C. R. Acad. Sci. Paris Sér. I Math. 292(1), 15–18 (1981)

    MathSciNet  MATH  Google Scholar 

  2. Bernstein, J.: Algebraic theory of D-modules (preprint)

  3. Baranovsky, V., Ginzburg, V. (personal communication)

  4. Bezrukavnikov, R., Kaledin, D.: Fedosov quantization in algebraic context. Mosc. Math. J. 4(3), 559–592 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bezrukavnikov, R., Losev, I.: Etingof conjecture for quantized quiver varieties, arXiv:1309.1716

  6. Bernstein, J., Lunts, V.: Equivariant sheaves and functors. Lecture Notes in Mathematics, vol. 1578. Springer-Verlag, Berlin (1994)

  7. Braden, T., Licata, A., Proudfoot, N., Webster, B.: Quantizations of conical symplectic resolutions II: category \(\cal{O}\) and symplectic duality. Astérisque 384, 75–179 (2016). (with an appendix by I. Losev)

    MathSciNet  MATH  Google Scholar 

  8. Braden, T., Proudfoot, N., Webster, B.: Quantizations of conical symplectic resolutions I: local and global structure. Astérisque 384, 1–73 (2016)

    MathSciNet  MATH  Google Scholar 

  9. Brundan, J.: On the definition of Kac-Moody 2-category. Math. Ann. 364(1–2), 353–372 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Crawley-Boevey, W., Etingof, P., Ginzburg, V.: Noncommutative geometry and quiver algebras. Adv. Math. 209(1), 274–336 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cautis, S., Dodd, C., Kamnitzer, J.: Associated graded of Hodge modules and categorical \(\mathfrak{sl}_2\) actions, arXiv:1603.07402

  12. Cautis, S., Kamnitzer, J.: Braiding via geometric Lie algebra actions. Compos. Math. 148(2), 464–506 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cautis, S., Kamnitzer, J., Licata, A.: Categorical geometric skew Howe duality. Invent. Math. 180(1), 111–159 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cautis, S., Kamnitzer, J., Licata, A.: Coherent sheaves on quiver varieties and categorification. Math. Ann. 357(3), 805–854 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cautis, S., Kamnitzer, J., Licata, A.: Derived equivalences for cotangent bundles of Grassmannians via categorical \(\mathfrak{sl}_2\) actions. J. Reine Angew. Math. 675, 53–99 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Cautis, S., Lauda, A.D.: Implicit structure in 2-representations of quantum groups. Selecta Math. (N.S.) 21(1), 201–244 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chuang, J., Rouquier, R.: Derived equivalences for symmetric groups and \(\mathfrak{sl}_2\)-categorification. Ann. Math. 167(1), 245–298 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. de Cataldo, M.A.A., Migliorini, L.: The decomposition theorem, perverse sheaves and the topology of algebraic maps. Bull. Am. Math. Soc. 46(4), 535–633 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Etingof, P., Gan, W.L., Ginzburg, V., Oblomkov, A.: Harish–Chandra homomorphisms and symplectic reflection algebras for wreath-products. Publ. Math. Inst. Hautes Études Sci. 105, 91–155 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Elias, B., Khovanov, M.: Diagrammatics for Soergel categories. Int. J. Math. Math. Sci. (2010). https://doi.org/10.1155/2010/978635

    Article  MathSciNet  MATH  Google Scholar 

  21. Etingof, P.: Symplectic reflection algebras and affine Lie algebras. Mosc. Math. J. 12(3), 543–565 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Elias, B., Williamson, G.: Soergel calculus. Represent. Theory 20, 295–374 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ginzburg, V.: Lectures on Nakajima’s quiver varieties, geometric methods in representation theory. In: I, Sémin. Congr., vol. 24, Soc. Math. France, Paris, pp. 145–219 (2012)

  24. Gordon, I.: A remark on rational Cherednik algebras and differential operators on the cyclic quiver. Glasg. Math. J. 48(1), 145–160 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hotta, R., Takeuchi, K., Tanisaki, T.: \(D\)-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, vol. 236. Birkhäuser Boston Inc., Boston (2008). (translated from the 1995 Japanese edition by Takeuchi)

    Book  MATH  Google Scholar 

  26. Khovanov, M., Lauda, A.D.: A categorification of quantum \({\mathfrak{s}l}(n)\). Quantum Topol. 1(1), 1–92 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kashiwara, M., Rouquier, R.: Microlocalization of rational Cherednik algebras. Duke Math. J. 144(3), 525–573 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kashiwara, M., Schapira, P.: Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 292. Springer-Verlag, Berlin (1994). (with a chapter in French by Christian Houzel, Corrected reprint of the 1990 original)

    Google Scholar 

  29. Kashiwara, M., Schapira, P.: Deformation quantization modules. Astérisque. 345, 147 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Kapustin, A., Witten, E.: Electric-magnetic duality and the geometric Langlands program. Commun. Number Theory Phys. 1(1), 1–236 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, Y.: On geometric realizations of quantum modified algebras and their canonical bases, arXiv:1007.5384

  32. Li, Y.: On geometric realizations of quantum modified algebras and their canonical bases II, arXiv:1009.0838

  33. Losev, I.: Isomorphisms of quantizations via quantization of resolutions. Adv. Math. 231, 1216–1270 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Am. Math. Soc. 4(2), 365–421 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  35. McGerty, K., Nevins, T.: Morse decomposition for D-module categories on stacks, arXiv:1402.7365

  36. Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76(2), 365–416 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  37. Nakajima, H.: Quiver varieties and Kac-Moody algebras. Duke Math. J. 91(3), 515–560 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  38. Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22(1), 233–286 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Rouquier, R.: 2-Kac-Moody algebras, arXiv:0812.5023

  40. Rouquier, R.: Quiver Hecke algebras and 2-Lie algebras. Algebra Colloq. 19(2), 359–410 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  42. Webster, B.: Centers of KLR algebras and cohomology rings of quiver varieties, arXiv:1504.04401

  43. Webster, B.: Unfurling Khovanov–Lauda–Rouquier algebras, arXiv:1603.06311

  44. Webster, B.: Weighted Khovanov–Lauda–Rouquier algebras, arXiv:1209.2463

  45. Webster, B.: Canonical bases and higher representation theory. Compos. Math. 151(1), 121–166 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  46. Webster, B.: Comparison of canonical bases for Schur and universal enveloping algebras. Transform. Groups 22(3), 869–883 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  47. Webster, B.: Knot invariants and higher representation theory. Mem. Am. Math. Soc. 250(1191), 141 (2017)

    MathSciNet  MATH  Google Scholar 

  48. Webster, B.: On generalized category \(\cal{O}\) for a quiver variety. Math. Ann. 368(1), 483–536 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wehrheim, K., Woodward, C.T.: Functoriality for Lagrangian correspondences in Floer theory. Quantum Topol. 1(2), 129–170 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  50. Zheng, H.: Categorification of integrable representations of quantum groups, arXiv:0803.3668

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Acknowledgements

This paper owes a great debt to Yiqiang Li; his work was an important inspiration, and he very helpfully pointed out a serious mistake in a draft version. The paper also benefited from many helpful comments by an anonymous referee. I also want to thank Nick Proudfoot, Tony Licata and Tom Braden; I depended very much on previous work and conversations with them to be able to write this paper. I thank Sabin Cautis and Aaron Lauda for sharing an early version of their paper with me. I also appreciate very stimulating conversations with Catharina Stroppel, Ivan Losev and Peter Tingley.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada

    Ben Webster

  2. Perimeter Institute for Mathematical Physics, Waterloo, ON, Canada

    Ben Webster

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  1. Ben Webster
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Correspondence to Ben Webster.

Additional information

Supported by the NSF under Grant DMS-1151473 and by the NSA under Grant H98230-10-1-0199. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

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Webster, B. A categorical action on quantized quiver varieties. Math. Z. 292, 611–639 (2019). https://doi.org/10.1007/s00209-018-2135-9

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  • DOI: https://doi.org/10.1007/s00209-018-2135-9

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