Selecta Mathematica

, 25:77 | Cite as

The circle quantum group and the infinite root stack of a curve

  • Francesco SalaEmail author
  • Olivier Schiffmann


In the present paper, we give a definition of the quantum group \(\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))\) of the circle \(S^1:={\mathbb {R}}/{\mathbb {Z}}\), and its fundamental representation. Such a definition is motivated by a realization of a quantum group \(\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))\) associated to the rational circle \(S^1_{\mathbb {Q}}:={\mathbb {Q}}/{\mathbb {Z}}\) as a direct limit of \(\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(n))\)’s, where the order is given by divisibility of positive integers. The quantum group \(\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))\) arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack \(X_\infty \) over a fixed smooth projective curve X defined over a finite field. Via this Hall algebra approach, we are able to realize geometrically the fundamental and the tensor representations, and a family of symmetric tensor representations, depending on the genus \(g_X\), of \(\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))\). Moreover, we show that \(\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(+\infty ))\) and \(\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(\infty ))\) are subalgebras of \(\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))\). As proved by T. Kuwagaki in the appendix, the quantum group \(\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))\) naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle \(S^1\).


Hall algebras Quantum groups Shuffle algebras Root stacks 

Mathematics Subject Classification

Primary: 17B37 17B67 22E65 Secondary: 14A20 



The authors are grateful to Mattia Talpo for many enlightening conversations about root stacks and their categories of coherent sheaves, to Jyun-Ao Lin for helpful explanations about his work and to the referee for a careful reading and useful comments. They also thank Andrea Appel and Mikhail Kapranov for helpful discussions and comments; and David Hernandez for pointing out his paper [25]. Last but not least, they thank Tatsuki Kuwagaki for writing the Appendix B. The author of the Appendix B thanks Francesco Sala for his intriguing talk at GTM seminar at IPMU, which motivated this note. He also thanks Francesco and Olivier Schiffmann for their kindness to include this note as an appendix of their work. The work of Tatsuki Kuwagaki was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan and JSPS KAKENHI Grant Number JP18K13405.


  1. 1.
    Abramovich, D., Graber, T., Vistoli, A.: Gromov–Witten theory of Deligne–Mumford stacks. Am. J. Math. 130(5), 1337–1398 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alper, J.: Good moduli spaces for Artin stacks. Ann. Inst. Fourier (Grenoble) 63(6), 2349–2402 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Appel, A., Sala, F.: Quantization of continuum Kac–Moody algebras. Pure Appl. Math. Q. (to appear) Google Scholar
  4. 4.
    Appel, A., Sala, F., Schiffmann, O.: Continuum Kac–Moody algebras. arXiv:1812.08528
  5. 5.
    Ariki, S., Jacon, N., Lecouvey, C.: Factorization of the canonical bases for higher-level Fock spaces. Proc. Edinb. Math. Soc. (2) 55(1), 23–51 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bayer, A., Cadman, C.: Quantum cohomology of \([\mathbb{C}^N/\mu _r]\). Compos. Math. 146(5), 1291–1322 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Beck, J.: Braid group action and quantum affine algebras. Commun. Math. Phys. 165(3), 555–568 (1994)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bergh, D., Lunts, V.A., Schnürer, O.M.: Geometricity for derived categories of algebraic stacks. Sel. Math. 22(4), 2535–2568 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bondal, A.I.: Derived categories of toric varieties. Convex and Algebraic geometry, Oberwolfach Conference Reports, vol. 3, pp. 284–286 (2006)Google Scholar
  10. 10.
    Borne, N.: Fibrés paraboliques et champ des racines. Int. Math. Res. Not. IMRN, no. 16, Art. ID rnm049, 38 (2007)Google Scholar
  11. 11.
    Borne, N., Vistoli, A.: Parabolic sheaves on logarithmic schemes. Adv. Math. 231(3–4), 1327–1363 (2012)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Bruzzo, U., Sala, F.: Framed sheaves on projective stacks. Adv. Math. 272, 20–95 (2015). (with an appendix by M. Pedrini) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Burban, I., Schiffmann, O.: On the Hall algebra of an elliptic curve, I. Duke Math. J. 161(7), 1171–1231 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Burban, I., Schiffmann, O.: The composition Hall algebra of a weighted projective line. J. Reine Angew. Math. 679, 75–124 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cadman, C.: Using stacks to impose tangency conditions on curves. Am. J. Math. 129(2), 405–427 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Canonaco, A.: The Beilinson complex and canonical rings of irregular surfaces. Mem. Am. Math. Soc. 183(862), viii+99 (2006)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Dou, R., Jiang, Y., Xiao, J.: The Hall algebra approach to Drinfeld’s presentation of quantum loop algebras. Adv. Math. 231(5), 2593–2625 (2012)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fang, B., Liu, C.-C.M., Treumann, D., Zaslow, E.: A categorification of Morelli’s theorem. Invent. Math. 186(1), 79–114 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fang, B., Liu, C.-C.M., Treumann, D., Zaslow, E.: The coherent–constructible correspondence for toric Deligne–Mumford stacks. Int. Math. Res. Not. IMRN 4, 914–954 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Fantechi, B., Mann, E., Nironi, F.: Smooth toric Deligne–Mumford stacks. J. Reine Angew. Math. 648, 201–244 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Ganatra, S., Pardon, J., Shende, V.: Covariantly functorial wrapped Floer theory on Liouville sectors. arXiv:1706.03152
  22. 22.
    Geigle, W., Lenzing, H.: A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), Lecture Notes in Math., vol. 1273, pp. 265–297. Springer, Berlin (1987)Google Scholar
  23. 23.
    Green, J.A.: Hall algebras, hereditary algebras and quantum groups. Invent. Math. 120(2), 361–377 (1995)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Hayashi, T.: \(q\)-analogues of Clifford and Weyl algebras—spinor and oscillator representations of quantum enveloping algebras. Commun. Math. Phys. 127(1), 129–144 (1990)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hernandez, D.: The algebra \(\cal{U}_q(\widehat{\mathfrak{sl}}_\infty )\) and applications. J. Algebra 329, 147–162 (2011)MathSciNetGoogle Scholar
  26. 26.
    Hori, K., Vafa, C.: Mirror symmetry. arXiv:hep-th/0002222
  27. 27.
    Hubery, A.: Symmetric functions and the centre of the Ringel–Hall algebra of a cyclic quiver. Math. Z. 251(3), 705–719 (2005)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kapranov, M.M.: Eisenstein series and quantum affine algebras. J. Math. Sci. (New York) 84(5), 1311–1360 (1997). (Algebraic geometry, 7) MathSciNetzbMATHGoogle Scholar
  29. 29.
    Kashiwara, M., Miwa, T., Stern, E.: Decomposition of \(q\)-deformed Fock spaces. Sel. Math. (N. S.) 1(4), 787–805 (1995)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften, vol. 292. Springer, Berlin (1990)Google Scholar
  31. 31.
    Keel, S., Mori, S.: Quotients by groupoids. Ann. Math. 145(1), 193–213 (1997)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Kontsevich, M.: Symplectic geometry of homological algebra.
  33. 33.
    Kuwagaki, T.: The nonequivariant coherent–constructible correspondence for toric stacks. arXiv:1610.03214
  34. 34.
    Laumon, G., Moret-Bailly, L.: Champs Algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39. Springer, Berlin (2000)Google Scholar
  35. 35.
    Lieblich, M.: Moduli of twisted sheaves. Duke Math. J. 138(1), 23–118 (2007)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Lin, J.-A.: Spherical Hall algebras of a weighted projective curve. arXiv:1410.0896
  37. 37.
    Mansuy, M.: Extremal loop weight modules for \(\cal{U}_q(\widehat{\mathfrak{sl}}_\infty )\). Algebras Represent. Theory 18(6), 1505–1532 (2015)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Mehta, V.B., Seshadri, C.S.: Moduli of vector bundles on curves with parabolic structures. Math. Ann. 248(3), 205–239 (1980)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Misra, K., Miwa, T.: Crystal base for the basic representation of \(\mathbf{U}_q(\hat{\mathfrak{sl}}(n))\). Commun. Math. Phys. 134(1), 79–88 (1990)zbMATHGoogle Scholar
  40. 40.
    Nadler, D.: Microlocal branes are constructible sheaves. Sel. Math. (N.S.) 15(4), 563–619 (2009)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22(1), 233–286 (2009)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Nironi, F.: Grothendieck Duality for Deligne–Mumford Stacks. arXiv:0811.1955
  43. 43.
    Nironi, F.: Moduli spaces of semistable sheaves on projective Deligne–Mumford stacks. arXiv:0811.1949
  44. 44.
    Olsson, M.: Logarithmic geometry and algebraic stacks. Ann. Sci. École Norm. Sup. (4) 36(5), 747–791 (2003)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Ringel, C.M.: Hall algebras and quantum groups. Invent. Math. 101(3), 583–591 (1990)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Ringel, C.M.: The composition algebra of a cyclic quiver. Towards an explicit description of the quantum group of type \(A_n^{(1)}\). Proc. Lond. Math. Soc. (3) 66(3), 507–537 (1993)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Sala, F., Schiffmann, O.: Fock space representation of the circle quantum group. Int. Math. Res. Not. IMRN (2019). Google Scholar
  48. 48.
    Schiffmann, O.: The Hall algebra of a cyclic quiver and canonical bases of Fock spaces. Int. Math. Res. Not. IMRN 8, 413–440 (2002)zbMATHGoogle Scholar
  49. 49.
    Schiffmann, O.: Noncommutative projective curves and quantum loop algebras. Duke Math. J. 121(1), 113–168 (2004)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Schiffmann, O.: Lectures on Hall algebras, Geometric methods in representation theory. II, Sémin. Congr., vol. 24, pp. 1–141. Soc. Math. France, Paris (2012)Google Scholar
  51. 51.
    Schiffmann, O., Vasserot, E.: Hall algebras of curves, commuting varieties and Langlands duality. Math. Ann. 353(4), 1399–1451 (2012)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Seidel, P.: Vanishing cycles and mutation, European Congress of Mathematics, Vol. II (Barcelona, 2000), Progr. Math., vol. 202, pp. 65–85. Birkhäuser, Basel (2001)Google Scholar
  53. 53.
    Talpo, M.: Infinite root stacks of logarithmic schemes and moduli of parabolic sheaves. Ph.D. thesis, Scuola Normale Superiore di Pisa, Italy (2015)Google Scholar
  54. 54.
    Talpo, M.: Moduli of parabolic sheaves on a polarized logarithmic scheme. Trans. Am. Math. Soc. 369(5), 3483–3545 (2017)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Talpo, M., Vistoli, A.: Infinite root stacks and quasi-coherent sheaves on logarithmic schemes. Proc. LMS 116(5), 1187–1243 (2018)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Xiao, J.: Drinfeld double and Ringel–Green theory of Hall algebra. J. Algebra 90, 100–144 (1997)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Kavli IPMU (WPI), UTIASThe University of TokyoKashiwa, ChibaJapan
  2. 2.Département de MathématiquesUniversité de Paris-Sud Paris-SaclayOrsay CedexFrance

Personalised recommendations