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TWO DESCRIPTIONS OF THE QUANTUM AFFINE ALGEBRA Uv() VIA HALL ALGEBRA APPROACH

Published online by Cambridge University Press:  12 December 2011

IGOR BURBAN  and
OLIVIER SCHIFFMANN
IGOR BURBAN
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany e-mail: [email protected]
OLIVIER SCHIFFMANN
Affiliation:
Département de Mathématiques Université Paris Sud Bâtiment 425 91405 Orsay CedexFrance e-mail: [email protected]
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Abstract

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We compare the reduced Drinfeld doubles of the composition subalgebras of the category of representations of the Kronecker quiver and the category of coherent sheaves on ℙ1. Using this approach, we show that the Drinfeld–Beck isomorphism for the quantized enveloping algebra Uv() is a corollary of an equivalence between the derived categories Db(Rep()) and Db(Coh(ℙ1)). This technique allows to reprove several results on the integral form of Uv().

Keywords

Primary 16T2018E30Secondary 17B37
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 2 , May 2012 , pp. 283 - 307
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras, Vol. 1: Techniques of representation theory (Cambridge University Press, Cambridge, UK, 2006).CrossRefGoogle Scholar
2.Baumann, P. and Kassel, C., The Hall algebra of the category of coherent sheaves on the projective line, J. Reine Angew. Math. 533 (2001), 207–233.Google Scholar
3.Beck, J., Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (3) (1994), 555–568.CrossRefGoogle Scholar
4.Beck, J., Chari, V. and Pressley, A., An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (3) (1999), 455–487.CrossRefGoogle Scholar
5.Beilinson, A., Coherent sheaves on ℙn and problems of linear algebra, Funct. Anal. Appl. 12 (1979), 214–216.CrossRefGoogle Scholar
6.Bernstein, J., Gelfand, I. and Ponomarev, A., Coxeter functors and Gabriel's theorem, Uspehi Mat. Nauk. 28 (2) (170) (1973), 19–33.Google Scholar
7.Bondal, A. and Kapranov, M., Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk. SSSR Ser. Mat. 53 (6) (1989), 1183–1205.Google Scholar
8.Bridgeland, T., Stability conditions on triangulated categories, Ann. Math. 166 (2) (2007), 317–345.CrossRefGoogle Scholar
9.Burban, I. and Schiffmann, O., On the Hall algebra of an elliptic curve I, arxiv: math.AG/0505148, Duke Math. J. (to appear).Google Scholar
10.Burban, I. and Schiffmann, O., Composition algebra of a weighted projective line, arXiv:1003.4412, J. Reine Angew. Math (to appear).Google Scholar
11.Chari, V. and Pressley, A., Quantum affine algebras at roots of unity, Represent. Theory 1 (1997), 280–328.CrossRefGoogle Scholar
12.Chari, V. and Pressley, A., A guide to quantum groups (Cambridge University Press, Cambridge, UK, 1994).Google Scholar
13.Cramer, T., Double Hall algebras and derived equivalences, Adv. Math. 224 (3) (2010), 1097–1120.CrossRefGoogle Scholar
14.Ding, J. and Frenkel, I., Isomorphism of two realizations of quantum affine algebra , Commun. Math. Phys. 156 (2) (1993), 277–300.CrossRefGoogle Scholar
15.Dold, A., Zur Homotopietheorie der Kettenkomplexe, Math. Ann. 140, (1960), 278–298.CrossRefGoogle Scholar
16.Drinfeld, V., A new realization of Yangians and of quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1) (1987), 13–17.Google Scholar
17.Gabriel, P., Auslander–Reiten sequences and representation-finite algebras, Representation theory I, proc. workshop, Ottawa 1979, Lect. Notes Math. 831 (1980), 1–71.CrossRefGoogle Scholar
18.Geigle, W. and Lenzing, H., A class of weighted projective curves arising in representation theory of finite dimensional algebras, singularities, representation of algebras and vector bundles, proc. symp., Lambrecht, 1985, Lect. Notes Math. 1273 (1987), 265–297.CrossRefGoogle Scholar
19.Green, J., Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (2), 361–377 (1995).CrossRefGoogle Scholar
20.Happel, D., Triangulated categories in the representation theory of finite dimensional algebras, LMS lecture note series, vol. 119 (Cambridge University Press, Cambridge, UK, 1988).Google Scholar
21.Hartshorne, R., Residues and duality, Lecture notes in mathematics, vol. 20 (Springer, Berlin, 1966).Google Scholar
22.Jimbo, M. and Miwa, T., Algebraic analysis of solvable lattice models, Regional conference series in mathematics, vol. 85 (American Mathematical Society, Providence, RI, 1995).Google Scholar
23.Jing, N., On Drinfeld realization of quantum affine algebras, the Monster and Lie algebras, in Ohio State University Mathematical Research Institute Publications (Ferrar, J. and Harada, K., Editors), vol. 7 (de Gruyter, Berlin, 1998), 195–206.Google Scholar
24.Joseph, A., Quantum groups and their primitive ideals (Springer, Berlin, 1995).CrossRefGoogle Scholar
25.Kapranov, M., Eisenstein series and quantum affine algebras, in Algebraic geometry, 7, J. Math. Sci. 84 (5) (1997), 1311–1360.CrossRefGoogle Scholar
26.Keller, B., Derived categories and their uses, in Handbook of algebra (Hazewinkel, M., Editor), vol. 1 (Elsevier, New York, 1996), 671–701.CrossRefGoogle Scholar
27.Lusztig, G., Introduction to quantum groups, Progress in mathematics, vol. 110 (Birkhäuser, Basel, Switzerland, 1993).Google Scholar
28.McGerty, K., The Kronecker quiver and bases of quantum affine , Adv. Math. 197 (2) (2005), 411–429.CrossRefGoogle Scholar
29.Peng, L. and Xiao, J., Root categories and simple Lie algebras, J. Algebra 198 (1) (1997), 19–56.CrossRefGoogle Scholar
30.Reineke, M., The quantum Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2) (2003), 349–368.CrossRefGoogle Scholar
31.Reiten, I. and Van den Bergh, M., Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2) (2002), 295–366.CrossRefGoogle Scholar
32.Ringel, C.-M., Tame algebras and quadratic forms, Lecture notes in mathematics, vol. 1099 (Springer, New York, 1984).Google Scholar
33.Ringel, C.-M., Hall algebras and quantum groups, Invent. Math. 101 (3) (1990), 583–591.CrossRefGoogle Scholar
34.Ringel, C.-M., PBW-bases of quantum groups, J. Reine Angew. Math. 470 (1996), 51–88.Google Scholar
35.Ringel, C.-M., Green's theorem on Hall algebras, CMS Conf. Proc. 19 (1996), 185–245.Google Scholar
36.Rudakov, A., Stability for an abelian category, J. Algebra 197 (1) (1997), 231–245.CrossRefGoogle Scholar
37.Schiffmann, O., Canonical bases and moduli spaces of sheaves on curves, Invent. Math. 165 (3) (2006), 453–524.CrossRefGoogle Scholar
38.Schiffmann, O., Geometric representation theory II (Brion, M., Editor), Séminaires et Congrès 25 (Société Mathématique de France, Paris, 2010), 1–141.Google Scholar
39.Sevenhant, B. and Van den Bergh, M., On the double of the Hall algebra of a quiver, J. Algebra 221 (1) (1999), 135–160.CrossRefGoogle Scholar
40.Szántó, C., Hall numbers and the composition algebra of the Kronecker algebra, Algebr. Represent. Theory 9 (5), 465–495 (2006).CrossRefGoogle Scholar
41.Tepetla, M., Coxeter functors: From their birth to tilting theory, Master Thesis (Norwegian University of Science and Technology, Trondheim, Norway, 2006).Google Scholar
42.Xiao, J., Drinfeld double and Ringel–Hall theory of Hall algebras, J. Algebra 190 (1) (1997), 100–144.CrossRefGoogle Scholar
43.Xiao, J. and Yang, S., BGP-reflection functors and Lusztig's symmetries: A Ringel–Hall algebra approach to quantum groups, J. Algebra 241 (1) (2001), 204–246.CrossRefGoogle Scholar
44.Xiao, J. and Zhang, G., A trip from representations of the Kronecker quiver to canonical bases of quantum affine algebras, representations of algebraic groups, quantum groups, and Lie algebras, in Contemporary mathematics, vol. 413 (American Mathematical Society, Providence, RI, 2006), 231–254.Google Scholar
45.Zhang, P., PBW-basis for the composition algebra of the Kronecker algebra, J. Reine Angew. Math. 527 (2000), 97–116.Google Scholar