Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 
 

 

Two-sided BGG resolutions of admissible representations


Author: Tomoyuki Arakawa
Journal: Represent. Theory 18 (2014), 183-222
MSC (2010): Primary 06B15, 17B67, 81R10
DOI: https://doi.org/10.1090/S1088-4165-2014-00454-0
Published electronically: August 7, 2014
MathSciNet review: 3244449
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the conjecture of Frenkel, Kac and Wakimoto on the existence of two-sided BGG resolutions of $ G$-integrable admissible representations of affine Kac-Moody algebras at fractional levels. As an application we establish the semi-infinite analogue of the generalized Borel-Weil theorem for minimal parabolic subalgebras which enables an inductive study of admissible representations.


References [Enhancements On Off] (What's this?)

  • [AG] S. Arkhipov and D. Gaitsgory, Differential operators on the loop group via chiral algebras, Int. Math. Res. Not. 4 (2002), 165-210. MR 1876958 (2003f:20077), https://doi.org/10.1155/S1073792802102078
  • [AL] H. H. Andersen and N. Lauritzen, Twisted Verma modules, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 1-26. MR 1985191 (2004d:17005)
  • [AM] Dražen Adamović and Antun Milas, Vertex operator algebras associated to modular invariant representations for $ A^{(1)}_1$, Math. Res. Lett. 2 (1995), no. 5, 563-575. MR 1359963 (96m:17047), https://doi.org/10.4310/MRL.1995.v2.n5.a4
  • [A1] Tomoyuki Arakawa, Vanishing of cohomology associated to quantized Drinfeld-Sokolov reduction, Int. Math. Res. Not. 15 (2004), 730-767. MR 2040965 (2005e:17033), https://doi.org/10.1155/S1073792804132479
  • [A2] Tomoyuki Arakawa, Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture, Duke Math. J. 130 (2005), no. 3, 435-478. MR 2184567 (2007a:17039), https://doi.org/10.1215/S0012-7094-05-13032-0
  • [A3] Tomoyuki Arakawa, Representation theory of $ \mathcal {W}$-algebras, Invent. Math. 169 (2007), no. 2, 219-320. MR 2318558 (2009d:17039), https://doi.org/10.1007/s00222-007-0046-1
  • [A4] Tomoyuki Arakawa, Representation theory of $ W$-algebras, II, Exploring new structures and natural constructions in mathematical physics, Adv. Stud. Pure Math., vol. 61, Math. Soc. Japan, Tokyo, 2011, pp. 51-90. MR 2867144
  • [A5] Tomoyuki Arakawa,
    Associated varieties of modules over Kac-Moody algebras and $ C_2$-cofiniteness of W-algebras.
    arXiv:1004.1554[math.QA].
  • [A6] T. Arakawa, Rationality of admissible affine vertex algebras in the category $ \mathcal {O}$, arXiv:1207.4857[math.QA].
  • [A7] Tomoyuki Arakawa, Rationality of $ \mathrm {W}$-algebras; principal nilpotent cases,
    arXiv:1211.7124[math.QA].
  • [Ark1] Sergey Arkhipov,
    A new construction of the semi-infinite BGG resolution.
    preprint, 1996.
    math.QA/9605043.
  • [Ark2] S. M. Arkhipov, Semi-infinite cohomology of associative algebras and bar duality, Internat. Math. Res. Notices 17 (1997), 833-863. MR 1474841 (98j:16006), https://doi.org/10.1155/S1073792897000548
  • [AS] Henning Haahr Andersen and Catharina Stroppel, Twisting functors on $ \mathcal {O}$, Represent. Theory 7 (2003), 681-699 (electronic). MR 2032059 (2004k:17010), https://doi.org/10.1090/S1088-4165-03-00189-4
  • [BF] D. Bernard and G. Felder, Fock representations and BRST cohomology in $ {\rm SL}(2)$ current algebra, Comm. Math. Phys. 127 (1990), no. 1, 145-168. MR 1036119 (91g:17022)
  • [BGG] I. N. Bernšteĭn, I. M. Gelfand, and S. I. Gelfand.
    Differential operators on the base affine space and a study of $ {\mathfrak{g}}$-modules.
    In Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971), pages 21-64. Halsted, New York, 1975.
  • [Feĭ] B. L. Feĭgin, Semi-infinite homology of Lie, Kac-Moody and Virasoro algebras, Uspekhi Mat. Nauk 39 (1984), no. 2(236), 195-196 (Russian). MR 740035 (85g:17003)
  • [FF1] B. L. Feĭgin and E. V. Frenkel, A family of representations of affine Lie algebras, Uspekhi Mat. Nauk 43 (1988), no. 5(263), 227-228 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 5, 221-222. MR 971497 (89k:17016), https://doi.org/10.1070/RM1988v043n05ABEH001935
  • [FF2] B. L. Feĭgin and E. V. Frenkel, Affine Kac-Moody algebras and semi-infinite flag manifolds, Comm. Math. Phys. 128 (1990), no. 1, 161-189. MR 1042449 (92f:17026)
  • [Fie] Peter Fiebig, The combinatorics of category $ \mathcal {O}$ over symmetrizable Kac-Moody algebras, Transform. Groups 11 (2006), no. 1, 29-49. MR 2205072 (2006k:17040), https://doi.org/10.1007/s00031-005-1103-8
  • [FKW] Edward Frenkel, Victor Kac, and Minoru Wakimoto, Characters and fusion rules for $ W$-algebras via quantized Drinfel'd-Sokolov reduction, Comm. Math. Phys. 147 (1992), no. 2, 295-328. MR 1174415 (93i:17029)
  • [FM] Igor Frenkel and Fyodor Malikov,
    Kazhdan-Lusztig tensoring and Harish-Chandra categories.
    preprint, 1997.
    arXiv:q-alg/9703010.
  • [Fre1] Edward Frenkel, Determinant formulas for the free field representations of the Virasoro and Kac-Moody algebras, Phys. Lett. B 286 (1992), no. 1-2, 71-77. MR 1175574 (93k:17056), https://doi.org/10.1016/0370-2693(92)90160-6
  • [Fre2] Edward Frenkel, Wakimoto modules, opers and the center at the critical level, Adv. Math. 195 (2005), no. 2, 297-404. MR 2146349 (2006d:17018), https://doi.org/10.1016/j.aim.2004.08.002
  • [GL] Howard Garland and James Lepowsky, Lie algebra homology and the Macdonald-Kac formulas, Invent. Math. 34 (1976), no. 1, 37-76. MR 0414645 (54 #2744)
  • [HT] Shinobu Hosono and Akihiro Tsuchiya, Lie algebra cohomology and $ N=2$ SCFT based on the GKO construction, Comm. Math. Phys. 136 (1991), no. 3, 451-486. MR 1099691 (92i:17042)
  • [Kos] Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329-387. MR 0142696 (26 #265)
  • [KRW] Victor Kac, Shi-Shyr Roan, and Minoru Wakimoto, Quantum reduction for affine superalgebras, Comm. Math. Phys. 241 (2003), no. 2-3, 307-342. MR 2013802 (2004h:17024)
  • [KT] Masaki Kashiwara and Toshiyuki Tanisaki, Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebras. III. Positive rational case, Asian J. Math. 2 (1998), no. 4, 779-832. Mikio Sato: a great Japanese mathematician of the twentieth century. MR 1734129 (2001f:17050)
  • [KW1] Victor G. Kac and Minoru Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 14, 4956-4960. MR 949675 (89j:17019), https://doi.org/10.1073/pnas.85.14.4956
  • [KW2] V. G. Kac and M. Wakimoto, Classification of modular invariant representations of affine algebras, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., vol. 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 138-177. MR 1026952 (91a:17032)
  • [KW3] Victor G. Kac and Minoru Wakimoto, On rationality of $ W$-algebras, Transform. Groups 13 (2008), no. 3-4, 671-713. MR 2452611 (2009i:17042), https://doi.org/10.1007/s00031-008-9028-7
  • [Lus] George Lusztig, Hecke algebras and Jantzen's generic decomposition patterns, Adv. in Math. 37 (1980), no. 2, 121-164. MR 591724 (82b:20059), https://doi.org/10.1016/0001-8708(80)90031-6
  • [MF] F. G. Malikov and I. B. Frenkel, Annihilating ideals and tilting functors, Funktsional. Anal. i Prilozhen. 33 (1999), no. 2, 31-42, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 33 (1999), no. 2, 106-115. MR 1719326 (2002c:17037), https://doi.org/10.1007/BF02465191
  • [Pet] D. Peterson,
    Quantum cohomology of $ G/P$.
    Lecture Notes, Cambridge, MA, Spring, Massachusetts Institute of Technology, 1997.
  • [RCW] Alvany Rocha-Caridi and Nolan R. Wallach, Projective modules over graded Lie algebras. I, Math. Z. 180 (1982), no. 2, 151-177. MR 661694 (83h:17018), https://doi.org/10.1007/BF01318901
  • [Soe1] Wolfgang Soergel, Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules, Represent. Theory 1 (1997), 83-114 (electronic). MR 1444322 (98d:17026), https://doi.org/10.1090/S1088-4165-97-00021-6
  • [Soe2] Wolfgang Soergel, Character formulas for tilting modules over Kac-Moody algebras, Represent. Theory 2 (1998), 432-448 (electronic). MR 1663141 (2000c:17048), https://doi.org/10.1090/S1088-4165-98-00057-0
  • [TK] Akihiro Tsuchiya and Yukihiro Kanie, Fock space representations of the Virasoro algebra. Intertwining operators, Publ. Res. Inst. Math. Sci. 22 (1986), no. 2, 259-327. MR 849260 (87k:17020), https://doi.org/10.2977/prims/1195178069
  • [Vor1] Alexander A. Voronov, Semi-infinite homological algebra, Invent. Math. 113 (1993), no. 1, 103-146. MR 1223226 (94f:17021), https://doi.org/10.1007/BF01244304
  • [Vor2] Alexander A. Voronov, Semi-infinite induction and Wakimoto modules, Amer. J. Math. 121 (1999), no. 5, 1079-1094. MR 1713301 (2000g:17025)
  • [Wak] Minoru Wakimoto, Fock representations of the affine Lie algebra $ A^{(1)}_1$, Comm. Math. Phys. 104 (1986), no. 4, 605-609. MR 841673 (87m:17011)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 06B15, 17B67, 81R10

Retrieve articles in all journals with MSC (2010): 06B15, 17B67, 81R10


Additional Information

Tomoyuki Arakawa
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 Japan
Email: arakawa@kurims.kyoto-u.ac.jp

DOI: https://doi.org/10.1090/S1088-4165-2014-00454-0
Received by editor(s): April 9, 2013
Received by editor(s) in revised form: April 10, 2013, and June 6, 2014
Published electronically: August 7, 2014
Additional Notes: This work was partially supported by JSPS KAKENHI Grant Number No. 20340007 and No. 23654006.
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society