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

BURBAN, IGOR ; SCHIFFMANN, OLIVIER

Glasgow mathematical journal, Tome 54 (2012) no. 2, pp. 283-307

Résumé

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().

Détail

DOI : 10.1017/S0017089511000607

Mots-clés : Primary 16T20, 18E30, Secondary 17B37

Bibliographie

[1] 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). Google Scholar | DOI

[2] 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] 3.Beck, J., Braid group action and quantum affine algebras, Comm. Math. Phys. 165 (3) (1994), 555–568. Google Scholar | DOI

[4] 4.Beck, J., Chari, V. and Pressley, A., An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (3) (1999), 455–487. Google Scholar | DOI

[5] 5.Beilinson, A., Coherent sheaves on ℙn and problems of linear algebra, Funct. Anal. Appl. 12 (1979), 214–216. Google Scholar | DOI

[6] 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] 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] 8.Bridgeland, T., Stability conditions on triangulated categories, Ann. Math. 166 (2) (2007), 317–345. Google Scholar | DOI

[9] 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] 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] 11.Chari, V. and Pressley, A., Quantum affine algebras at roots of unity, Represent. Theory 1 (1997), 280–328. Google Scholar | DOI

[12] 12.Chari, V. and Pressley, A., A guide to quantum groups (Cambridge University Press, Cambridge, UK, 1994). Google Scholar

[13] 13.Cramer, T., Double Hall algebras and derived equivalences, Adv. Math. 224 (3) (2010), 1097–1120. Google Scholar | DOI

[14] 14.Ding, J. and Frenkel, I., Isomorphism of two realizations of quantum affine algebra , Commun. Math. Phys. 156 (2) (1993), 277–300. Google Scholar | DOI

[15] 15.Dold, A., Zur Homotopietheorie der Kettenkomplexe, Math. Ann. 140, (1960), 278–298. Google Scholar | DOI

[16] 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] 17.Gabriel, P., Auslander–Reiten sequences and representation-finite algebras, Representation theory I, proc. workshop, Ottawa 1979, Lect. Notes Math. 831 (1980), 1–71. Google Scholar | DOI

[18] 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. Google Scholar | DOI

[19] 19.Green, J., Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (2), 361–377 (1995). Google Scholar | DOI

[20] 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] 21.Hartshorne, R., Residues and duality, Lecture notes in mathematics, vol. 20 (Springer, Berlin, 1966). Google Scholar

[22] 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] 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] 24.Joseph, A., Quantum groups and their primitive ideals (Springer, Berlin, 1995). Google Scholar | DOI

[25] 25.Kapranov, M., Eisenstein series and quantum affine algebras, in Algebraic geometry, 7, J. Math. Sci. 84 (5) (1997), 1311–1360. Google Scholar | DOI

[26] 26.Keller, B., Derived categories and their uses, in Handbook of algebra (Hazewinkel, M., Editor), vol. 1 (Elsevier, New York, 1996), 671–701. Google Scholar | DOI

[27] 27.Lusztig, G., Introduction to quantum groups, Progress in mathematics, vol. 110 (Birkhäuser, Basel, Switzerland, 1993). Google Scholar

[28] 28.McGerty, K., The Kronecker quiver and bases of quantum affine , Adv. Math. 197 (2) (2005), 411–429. Google Scholar | DOI

[29] 29.Peng, L. and Xiao, J., Root categories and simple Lie algebras, J. Algebra 198 (1) (1997), 19–56. Google Scholar | DOI

[30] 30.Reineke, M., The quantum Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2) (2003), 349–368. Google Scholar | DOI

[31] 31.Reiten, I. and Van den Bergh, M., Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2) (2002), 295–366. Google Scholar | DOI

[32] 32.Ringel, C.-M., Tame algebras and quadratic forms, Lecture notes in mathematics, vol. 1099 (Springer, New York, 1984). Google Scholar

[33] 33.Ringel, C.-M., Hall algebras and quantum groups, Invent. Math. 101 (3) (1990), 583–591. Google Scholar | DOI

[34] 34.Ringel, C.-M., PBW-bases of quantum groups, J. Reine Angew. Math. 470 (1996), 51–88. Google Scholar

[35] 35.Ringel, C.-M., Green's theorem on Hall algebras, CMS Conf. Proc. 19 (1996), 185–245. Google Scholar

[36] 36.Rudakov, A., Stability for an abelian category, J. Algebra 197 (1) (1997), 231–245. Google Scholar | DOI

[37] 37.Schiffmann, O., Canonical bases and moduli spaces of sheaves on curves, Invent. Math. 165 (3) (2006), 453–524. Google Scholar | DOI

[38] 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] 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. Google Scholar | DOI

[40] 40.Szántó, C., Hall numbers and the composition algebra of the Kronecker algebra, Algebr. Represent. Theory 9 (5), 465–495 (2006). Google Scholar | DOI

[41] 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] 42.Xiao, J., Drinfeld double and Ringel–Hall theory of Hall algebras, J. Algebra 190 (1) (1997), 100–144. Google Scholar | DOI

[43] 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. Google Scholar | DOI

[44] 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] 45.Zhang, P., PBW-basis for the composition algebra of the Kronecker algebra, J. Reine Angew. Math. 527 (2000), 97–116. Google Scholar