Quantum groups via Hall algebras of complexes

Abstract

We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a finite field Hall algebra construction involving $\mathbb{Z}_2$-graded complexes of quiver representations.

  • [burb_schiff] Go to document I. Burban and O. Schiffmann, "On the Hall algebra of an elliptic curve, I," Duke Math. J., vol. 161, iss. 7, pp. 1171-1231, 2012.
    @article {burb_schiff, MRKEY = {2922373},
      AUTHOR = {Burban, Igor and Schiffmann, Olivier},
      TITLE = {On the {H}all algebra of an elliptic curve, {I}},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {161},
      YEAR = {2012},
      NUMBER = {7},
      PAGES = {1171--1231},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {16T05 (14Hxx 22Exx)},
      MRNUMBER = {2922373},
      DOI = {10.1215/00127094-1593263},
      ZBLNUMBER = {06047809},
      }
  • [burban] Go to document I. Burban and O. Schiffmann, "Two descriptions of the quantum affine algebra $U_v(\widehat{\mathfrak{sl}}_2)$ via Hall algebra approach," Glasg. Math. J., vol. 54, iss. 2, pp. 283-307, 2012.
    @article {burban, MRKEY = {2911369},
      AUTHOR = {Burban, Igor and Schiffmann, Olivier},
      TITLE = {Two descriptions of the quantum affine algebra {$U\sb v(\widehat{\mathfrak{sl}}\sb 2)$} via {H}all algebra approach},
      JOURNAL = {Glasg. Math. J.},
      FJOURNAL = {Glasgow Mathematical Journal},
      VOLUME = {54},
      YEAR = {2012},
      NUMBER = {2},
      PAGES = {283--307},
      ISSN = {0017-0895},
      MRCLASS = {16G20 (14F05 17B37)},
      MRNUMBER = {2911369},
      DOI = {10.1017/S0017089511000607},
      ZBLNUMBER = {06031012},
      }
  • [cramer] Go to document T. Cramer, "Double Hall algebras and derived equivalences," Adv. Math., vol. 224, iss. 3, pp. 1097-1120, 2010.
    @article {cramer, MRKEY = {2628805},
      AUTHOR = {Cramer, Tim},
      TITLE = {Double {H}all algebras and derived equivalences},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {224},
      YEAR = {2010},
      NUMBER = {3},
      PAGES = {1097--1120},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {16G20 (16E35 17B37)},
      MRNUMBER = {2628805},
      MRREVIEWER = {Csaba Sz{á}nt{ó}},
      DOI = {10.1016/j.aim.2009.12.021},
      ZBLNUMBER = {1208.16016},
      }
  • [drinfeld] V. G. Drinfel$’$d, "Quantum groups," in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Providence, RI, 1987, pp. 798-820.
    @inproceedings {drinfeld, MRKEY = {0934283},
      AUTHOR = {Drinfel$'$d, V. G.},
      TITLE = {Quantum groups},
      BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2},
      VENUE={{B}erkeley, {C}alif., 1986},
      PAGES = {798--820},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1987},
      MRCLASS = {17B50 (16A24 17B65 57T05 58F07 82A05 82A15)},
      MRNUMBER = {0934283},
      }
  • [green] Go to document J. A. Green, "Hall algebras, hereditary algebras and quantum groups," Invent. Math., vol. 120, iss. 2, pp. 361-377, 1995.
    @article {green, MRKEY = {1329046},
      AUTHOR = {Green, James A.},
      TITLE = {Hall algebras, hereditary algebras and quantum groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {120},
      YEAR = {1995},
      NUMBER = {2},
      PAGES = {361--377},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {16G10 (16E30 16G20 16W30 16W50 17B37)},
      MRNUMBER = {1329046},
      MRREVIEWER = {C. M. Ringel},
      DOI = {10.1007/BF01241133},
      ZBLNUMBER = {0836.16021},
      }
  • [joseph] A. Joseph, Quantum groups and their primitive ideals, New York: Springer-Verlag, 1995.
    @book {joseph, MRKEY = {1315966},
      AUTHOR = {Joseph, Anthony},
      TITLE = {Quantum groups and their primitive ideals},
      SERIES = {Ergeb. Math. Grenzgeb.},
      NUMBER = {29},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1995},
      PAGES = {x+383},
      ISBN = {3-540-57057-8},
      MRCLASS = {17B37 (16-02 16W30 17-02 81R50)},
      MRNUMBER = {1315966},
      MRREVIEWER = {Kenneth A. Brown},
      ZBLNUMBER = {0808.17004},
      }
  • [kapranov] Go to document M. Kapranov, "Heisenberg doubles and derived categories," J. Algebra, vol. 202, iss. 2, pp. 712-744, 1998.
    @article {kapranov, MRKEY = {1617651},
      AUTHOR = {Kapranov, M.},
      TITLE = {Heisenberg doubles and derived categories},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {202},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {712--744},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {16W30 (18E30)},
      MRNUMBER = {1617651},
      MRREVIEWER = {Dieter Happel},
      DOI = {10.1006/jabr.1997.7323},
      ZBLNUMBER = {0910.18005},
      }
  • [lusztig] G. Lusztig, Introduction to Quantum Groups, Boston, MA: Birkhäuser, 1993, vol. 110.
    @book {lusztig, MRKEY = {1227098},
      AUTHOR = {Lusztig, George},
      TITLE = {Introduction to Quantum Groups},
      SERIES = {Progr. Math.},
      VOLUME = {110},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {1993},
      PAGES = {xii+341},
      ISBN = {0-8176-3712-5},
      MRCLASS = {17B37 (16W30 17-02 17B35 81R50)},
      MRNUMBER = {1227098},
      MRREVIEWER = {Jie Du},
      ZBLNUMBER = {0788.17010},
      }
  • [peng_xiao] Go to document L. Peng and J. Xiao, "Root categories and simple Lie algebras," J. Algebra, vol. 198, iss. 1, pp. 19-56, 1997.
    @article {peng_xiao, MRKEY = {1482975},
      AUTHOR = {Peng, Liangang and Xiao, Jie},
      TITLE = {Root categories and simple {L}ie algebras},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {198},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {19--56},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {17B20 (16G20 18E30)},
      MRNUMBER = {1482975},
      MRREVIEWER = {Dieter Happel},
      DOI = {10.1006/jabr.1997.7152},
      ZBLNUMBER = {0893.16007},
      }
  • [peng_xiao2] Go to document L. Peng and J. Xiao, "Triangulated categories and Kac-Moody algebras," Invent. Math., vol. 140, iss. 3, pp. 563-603, 2000.
    @article {peng_xiao2, MRKEY = {1760751},
      AUTHOR = {Peng, Liangang and Xiao, Jie},
      TITLE = {Triangulated categories and {K}ac-{M}oody algebras},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {140},
      YEAR = {2000},
      NUMBER = {3},
      PAGES = {563--603},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {17B67 (16E60 16G10 18E30)},
      MRNUMBER = {1760751},
      MRREVIEWER = {James A. Green},
      DOI = {10.1007/s002220000062},
      ZBLNUMBER = {0966.16006},
     }
  • [ringel] Go to document C. M. Ringel, "Hall algebras and quantum groups," Invent. Math., vol. 101, iss. 3, pp. 583-591, 1990.
    @article {ringel, MRKEY = {1062796},
      AUTHOR = {Ringel, Claus Michael},
      TITLE = {Hall algebras and quantum groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {101},
      YEAR = {1990},
      NUMBER = {3},
      PAGES = {583--591},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {16E60 (16G60 16P10 16S30 17B37)},
      MRNUMBER = {1062796},
      MRREVIEWER = {Jie Du},
      DOI = {10.1007/BF01231516},
      ZBLNUMBER = {0735.16009},
      }
  • [ringel3] C. M. Ringel, "Green’s theorem on Hall algebras," in Representation Theory of Algebras and Related Topics, Providence, RI: Amer. Math. Soc., 1996, vol. 19, pp. 185-245.
    @incollection {ringel3, MRKEY = {1388564},
      AUTHOR = {Ringel, Claus Michael},
      TITLE = {Green's theorem on {H}all algebras},
      BOOKTITLE = {Representation Theory of Algebras and Related Topics},
      VENUE={{M}exico {C}ity, 1994},
      SERIES = {CMS Conf. Proc.},
      VOLUME = {19},
      PAGES = {185--245},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1996},
      MRCLASS = {16G10 (05A19 16W30)},
      MRNUMBER = {1388564},
      MRREVIEWER = {Jin Yun Guo},
      ZBLNUMBER = {0860.16026},
      }
  • [schiffmann] O. Schiffmann, Lectures on Hall algebras.
    @misc{schiffmann,
      author={Schiffmann, O.},
      TITLE={Lectures on {H}all algebras},
      ARXIV= {math/0611617},
     }
  • [toen] Go to document B. Toën, "Derived Hall algebras," Duke Math. J., vol. 135, iss. 3, pp. 587-615, 2006.
    @article {toen, MRKEY = {2272977},
      AUTHOR = {To{ë}n, Bertrand},
      TITLE = {Derived {H}all algebras},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {135},
      YEAR = {2006},
      NUMBER = {3},
      PAGES = {587--615},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {18G55 (18E30)},
      MRNUMBER = {2272977},
      MRREVIEWER = {David A. Blanc},
      DOI = {10.1215/S0012-7094-06-13536-6},
      ZBLNUMBER = {1117.18011},
      }
  • [xiao] Go to document J. Xiao, "Drinfeld double and Ringel–Green theory of Hall algebras," J. Algebra, vol. 190, iss. 1, pp. 100-144, 1997.
    @article {xiao, MRKEY = {1442148},
      AUTHOR = {Xiao, Jie},
      TITLE = {Drinfeld double and {R}ingel--{G}reen theory of {H}all algebras},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {190},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {100--144},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {16G20 (16W30 17B37 81R50)},
      MRNUMBER = {1442148},
      MRREVIEWER = {Steffen K{ö}nig},
      DOI = {10.1006/jabr.1996.6887},
      ZBLNUMBER = {0874.16026},
      }
  • [xiao_xu] Go to document J. Xiao and F. Xu, "Hall algebras associated to triangulated categories," Duke Math. J., vol. 143, iss. 2, pp. 357-373, 2008.
    @article {xiao_xu, MRKEY = {2420510},
      AUTHOR = {Xiao, Jie and Xu, Fan},
      TITLE = {Hall algebras associated to triangulated categories},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {143},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {357--373},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {18E30},
      MRNUMBER = {2420510},
      MRREVIEWER = {Sunil K. Chebolu},
      DOI = {10.1215/00127094-2008-021},
      ZBLNUMBER = {1168.18006},
      }
  • [yanagida] S. Yanagida, A note on Bridgeland’s Hall algebra of two-periodic complexes.
    @misc{yanagida,
      author={Yanagida, S.},
      TITLE={A note on {B}ridgeland's {H}all algebra of two-periodic complexes},
      ARXIV={1207.0905},
     }

Authors

Tom Bridgeland

All Souls College
Oxford OX1 4AL
U.K.