Duke Mathematical Journal

Noncommutative projective curves and quantum loop algebras

Olivier Schiffmann

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We show that the Hall algebra of the category of coherent sheaves on a weighted projective line over a finite field provides a realization of the (quantized) enveloping algebra of a certain nilpotent subalgebra of the affinization of the corresponding Kac-Moody algebra. In particular, this yields a geometric realization of the quantized enveloping algebra of elliptic (or $2$-toroidal) algebras of types $D_4^{(1,1)}$, $E^{(1,1)}_6$, $E^{(1,1)}_7$, and $E_{8}^{(1,1)}$ in terms of coherent sheaves on weighted projective lines of genus one or, equivalently, in terms of equivariant coherent sheaves on elliptic curves.

Article information

Duke Math. J., Volume 121, Number 1 (2004), 113-168.

First available in Project Euclid: 21 December 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E
Secondary: 16G 18F


Schiffmann, Olivier. Noncommutative projective curves and quantum loop algebras. Duke Math. J. 121 (2004), no. 1, 113--168. doi:10.1215/S0012-7094-04-12114-1. https://projecteuclid.org/euclid.dmj/1072058751

Export citation


  • P. Baumann and C. Kassel, The Hall algebra of the category of coherent sheaves on the projective line, J. Reine Angew. Math. 533 (2001), 207--233.
  • J. Beck, Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (1994), 555--568.
  • A. I. Bondal, Representations of associative algebras and coherent sheaves (in Russian), Izv. Akad. Nauk. SSSR Ser. Mat. 53, no. 1 (1989), 25--44.; English translation in Math. USSR-Izv. 34 (1990), 23--42.
  • B. Crawley-Boevey, Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity, preprint.
  • W. Crawley-Boevey and M. Van den Bergh, Absolutely indecomposable representations and Kac-Moody Lie algebras, appendix by H. Nakajima, to appear in Invent. Math..
  • B. Enriquez, PBW and duality theorems for quantum groups and quantum current algebras, J. Lie Theory 13 (2003), 21--64.
  • W. Geigle and H. Lenzing, ``A class of weighted projective curves arising in the representation theory of finite-dimensional algebras'' in Singularities, Representations of Algebras and Vector Bundles (Lambrecht, Germany, 1985), Lecture Notes in Math. 1273, Springer, Berlin, 1987, 265--297.
  • P. Hall, ``The algebra of partitions'' in Proceedings of the Fourth Canadian Mathematical Congress (Banff, Canada, 1957), Univ. of Toronto Press, Toronto, 1959, 147--159.
  • D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Math. Soc. Lecture Note Ser. 119, Cambridge Univ. Press., Cambridge, 1988.
  • --. --. --. --., A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), 381--398.
  • V. G. Kac, Infinite-Dimensional Lie Algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990.
  • M. M. Kapranov, ``Eisenstein series and quantum affine algebras'' in Algebraic Geometry, 7, J. Math. Sci. (New York) 84, Consultants Bureau, New York, 1997, 1311--1360.
  • K. Lamotke, Regular Solids and Isolated Singularities, Adv. Lectures Math., Vieweg, Braunschweig, Germany, 1986.
  • B. Leclerc and J.-Y. Thibon, ``Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials'' in Combinatorial Methods in Representation Theory (Kyoto, 1998), Adv. Stud. Pure Math. 28, Kinokuniya, Tokyo, 2000, 155--220.
  • H. Lenzing, ``Curve singularities arising from the representation theory of tame hereditary algebras'' in Representation Theory, I (Ottawa, Canada, 1984), Lecture Notes in Math. 1177, Springer, Berlin, 1986, 199--231.
  • H. Lenzing and H. Meltzer, ``Sheaves on a weighted projective line of genus one, and representations of a tubular algebra'' in Representations of Algebras (Ottawa, Canada, 1992), CMS Conf. Proc. 14, Amer. Math. Soc., Providence, 313--337.
  • Y. Lin and L. Peng, $2$-extended affine Lie algebras and tubular algebras, to appear in Adv. Math.
  • G. Lusztig, Introduction to Quantum Groups, Progr. Math. 110, Birkhäuser, Boston, 1993.
  • I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Math. Monogr., Oxford Univ. Press, New York, 1979.
  • R. V. Moody, S. E. Rao, and T. Yokonuma, Toroidal Lie algebras and vertex representations, Geom. Dedicata 35 (1990), 283--307.
  • H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math. J. 91 (1998), 515--560.
  • L. Peng and J. Xiao, Triangulated categories and Kac-Moody algebras, Invent. Math. 140 (2000), 563--603.
  • M. Reineke, The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2003), 349--368.
  • C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984.
  • --. --. --. --., Hall algebras and quantum groups, Invent. Math. 101 (1990), 583--591.
  • --. --. --. --., ``Hall algebras'' in Topics in Algebra, Part 1 (Warsaw, 1988), Banach Center Publ. 26, PWN, Warsaw, 1990, 433--447.
  • O. Schiffmann, The Hall algebra of a cyclic quiver and canonical bases of Fock spaces, Internat. Math. Res. Notices 2000, no. 8, 413--440.