Rationality of $W$-algebras: principal nilpotent cases

Abstract

We prove the rationality of all the minimal series principal $W$-algebras discovered by Frenkel, Kac and Wakimoto, thereby giving a new family of rational and $C_2$-cofinite vertex operator algebras. A key ingredient in our proof is the study of Zhu’s algebra of simple $W$-algebras via the quantized Drinfeld-Sokolov reduction. We show that the functor of taking Zhu’s algebra commutes with the reduction functor. Using this general fact we determine the maximal spectrums of the associated graded of Zhu’s algebras of vertex operator algebras associated with admissible representations of affine Kac-Moody algebras as well.

  • [AbeBuhDon04] Go to document T. Abe, G. Buhl, and C. Dong, "Rationality, regularity, and $C_2$-cofiniteness," Trans. Amer. Math. Soc., vol. 356, iss. 8, pp. 3391-3402, 2004.
    @article{AbeBuhDon04, mrkey = {2052955},
      author = {Abe, Toshiyuki and Buhl, Geoffrey and Dong, Chongying},
      title = {Rationality, regularity, and {$C\sb 2$}-cofiniteness},
      journal = {Trans. Amer. Math. Soc.},
      fjournal = {Transactions of the American Mathematical Society},
      volume = {356},
      year = {2004},
      number = {8},
      pages = {3391--3402},
      issn = {0002-9947},
      coden = {TAMTAM},
      mrclass = {17B69 (17B10)},
      mrnumber = {2052955},
      mrreviewer = {S. Eswara Rao},
      doi = {10.1090/S0002-9947-03-03413-5},
      zblnumber = {1070.17011},
      }
  • [AKM] Go to document T. Arakawa, T. Kuwabara, and F. Malikov, "Localization of affine W-algebras," Comm. Math. Phys., vol. 335, iss. 1, pp. 143-182, 2015.
    @article{AKM, MRKEY={3314502},
      AUTHOR = {Arakawa, T. and Kuwabara, T. and Malikov, F.},
      TITLE = {Localization of affine {W}-algebras},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {335},
      YEAR = {2015},
      NUMBER = {1},
      PAGES = {143--182},
      ISSN = {0010-3616},
      MRCLASS = {Preliminary Data},
      MRNUMBER = {3314502},
      zblnumber = {06413804},
      doi = {10.1007/s00220-014-2183-x},
     }
  • [ALY] Go to document T. Arakawa, C. H. Lam, and H. Yamada, "Zhu’s algebra, $C_2$-algebra and $C_2$-cofiniteness of parafermion vertex operator algebras," Adv. Math., vol. 264, pp. 261-295, 2014.
    @article{ALY, mrkey = {3250285},
      author = {Arakawa, Tomoyuki and Lam, Ching Hung and Yamada, Hiromichi},
      title = {Zhu's algebra, {$C\sb 2$}-algebra and {$C\sb 2$}-cofiniteness of parafermion vertex operator algebras},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {264},
      year = {2014},
      pages = {261--295},
      issn = {0001-8708},
      mrclass = {17B69},
      mrnumber = {3250285},
      doi = {10.1016/j.aim.2014.07.021},
      zblnumber = {06334304},
      }
  • [AdaMil95] Go to document D. Adamović and A. Milas, "Vertex operator algebras associated to modular invariant representations for $A^{(1)}_1$," Math. Res. Lett., vol. 2, iss. 5, pp. 563-575, 1995.
    @article{AdaMil95, mrkey = {1359963},
      author = {Adamovi{ć},
      Dra{ž}en and Milas, Antun},
      title = {Vertex operator algebras associated to modular invariant representations for {$A^{(1)}_1$}},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {2},
      year = {1995},
      number = {5},
      pages = {563--575},
      issn = {1073-2780},
      mrclass = {17B69 (17B10 17B67)},
      mrnumber = {1359963},
      mrreviewer = {Chong Ying Dong},
      doi = {10.4310/MRL.1995.v2.n5.a4},
      zblnumber = {0848.17033},
      }
  • [Ara05] Go to document T. Arakawa, "Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture," Duke Math. J., vol. 130, iss. 3, pp. 435-478, 2005.
    @article{Ara05, mrkey = {2184567},
      author = {Arakawa, Tomoyuki},
      title = {Representation theory of superconformal algebras and the {K}ac-{R}oan-{W}akimoto conjecture},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {130},
      year = {2005},
      number = {3},
      pages = {435--478},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {17B68 (17B10)},
      mrnumber = {2184567},
      mrreviewer = {Matthias D{ö}rrzapf},
      doi = {10.1215/S0012-7094-05-13032-0},
      zblnumber = {1112.17026},
      }
  • [Ara07] Go to document T. Arakawa, "Representation theory of $\mathcal{W}$-algebras," Invent. Math., vol. 169, iss. 2, pp. 219-320, 2007.
    @article{Ara07, mrkey = {2318558},
      author = {Arakawa, Tomoyuki},
      title = {Representation theory of {$\mathcal{W}$}-algebras},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {169},
      year = {2007},
      number = {2},
      pages = {219--320},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {17B68 (81R10)},
      mrnumber = {2318558},
      doi = {10.1007/s00222-007-0046-1},
      zblnumber = {1172.17019},
      }
  • [Ara08-a] T. Arakawa, "Representation theory of $W$-algebras, II," in Exploring New Structures and Natural Constructions in Mathematical Physics, Tokyo: Math. Soc. Japan, 2011, vol. 61, pp. 51-90.
    @incollection{Ara08-a, mrkey = {2867144},
      author = {Arakawa, Tomoyuki},
      title = {Representation theory of {$W$}-algebras, {II}},
      booktitle = {Exploring New Structures and Natural Constructions in Mathematical Physics},
      series = {Adv. Stud. Pure Math.},
      volume = {61},
      pages = {51--90},
      publisher = {Math. Soc. Japan},
      address = {Tokyo},
      year = {2011},
      mrclass = {17B68 (81R10)},
      mrnumber = {2867144},
      mrreviewer = {Panagiotis Batakidis},
      zblnumber = {1262.17014},
      }
  • [Ara12] Go to document T. Arakawa, "A remark on the $C_2$-cofiniteness condition on vertex algebras," Math. Z., vol. 270, iss. 1-2, pp. 559-575, 2012.
    @article{Ara12, mrkey = {2875849},
      author = {Arakawa, Tomoyuki},
      title = {A remark on the {$C\sb 2$}-cofiniteness condition on vertex algebras},
      journal = {Math. Z.},
      fjournal = {Mathematische Zeitschrift},
      volume = {270},
      year = {2012},
      number = {1-2},
      pages = {559--575},
      issn = {0025-5874},
      coden = {MAZEAX},
      mrclass = {17B69 (17B68)},
      mrnumber = {2875849},
      mrreviewer = {Hai Sheng Li},
      doi = {10.1007/s00209-010-0812-4},
      zblnumber = {06006390},
      }
  • [Ara09b] Go to document T. Arakawa, "Associated varieties of modules over Kac-Moody algebras and $C_2$-cofiniteness of W-algebras," Int. Math. Res. Not., vol. 2015.
    @article{Ara09b, KEY={Ara15},
      mrkey = {2040965},
      author = {Arakawa, Tomoyuki},
      title = {Associated varieties of modules over {K}ac-{M}oody algebras and {$C_2$}-cofiniteness of {W}-algebras},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices},
      SORTYEAR={2015},
      VOLUME = {2015},
      NOTE = {published online February 19, 2015, 62 pgs.},
      DOI={10.1093/imrn/rnu277 },
      }
  • [AraBP] Go to document T. Arakawa, "Rationality of Bershadsky-Polyakov vertex algebras," Comm. Math. Phys., vol. 323, iss. 2, pp. 627-633, 2013.
    @article{AraBP, mrkey = {3096533},
      author = {Arakawa, Tomoyuki},
      title = {Rationality of {B}ershadsky-{P}olyakov vertex algebras},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {323},
      year = {2013},
      number = {2},
      pages = {627--633},
      issn = {0010-3616},
      mrclass = {17B69 (81T40)},
      mrnumber = {3096533},
      mrreviewer = {Domenico Fiorenza},
      doi = {10.1007/s00220-013-1780-4},
      zblnumber = {06228489},
      }
  • [A-BGG] Go to document T. Arakawa, "Two-sided BGG resolutions of admissible representations," Represent. Theory, vol. 18, pp. 183-222, 2014.
    @article{A-BGG, mrkey = {3244449},
      author = {Arakawa, Tomoyuki},
      title = {Two-sided {BGG} resolutions of admissible representations},
      journal = {Represent. Theory},
      fjournal = {Representation Theory. An Electronic Journal of the American Mathematical Society},
      volume = {18},
      year = {2014},
      pages = {183--222},
      issn = {1088-4165},
      mrclass = {17B67 (17B55)},
      mrnumber = {3244449},
      mrreviewer = {Elizabeth Graf Jurisich},
      doi = {10.1090/S1088-4165-2014-00454-0},
      zblnumber = {06355678},
      }
  • [A12-2] T. Arakawa, Rationality of admissible affine vertex algebras in the category $\mathcal{O}$, 2012.
    @misc{A12-2,
      author = {Arakawa, Tomoyuki},
      title = {Rationality of admissible affine vertex algebras in the category {$\mathcal{O}$}},
      arxiv = {1207.4857},
      year = {2012},
      note = {to appear in {\it Duke Math. J.}},
      }
  • [BeiFeiMaz] A. Beilinson, B. Feigin, and B. Mazur, Introduction to algebraic field theory on curves.
    @misc{BeiFeiMaz,
      author = {Beilinson, A. and Feigin, B. and Mazur, B.},
      title = {Introduction to algebraic field theory on curves},
      note = {preprint},
      }
  • [BPZ84] Go to document A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, "Infinite conformal symmetry in two-dimensional quantum field theory," Nuclear Phys. B, vol. 241, iss. 2, pp. 333-380, 1984.
    @article{BPZ84, mrkey = {0757857},
      author = {Belavin, A. A. and Polyakov, A. M. and Zamolodchikov, A. B.},
      title = {Infinite conformal symmetry in two-dimensional quantum field theory},
      journal = {Nuclear Phys. B},
      fjournal = {Nuclear Physics. B},
      volume = {241},
      year = {1984},
      number = {2},
      pages = {333--380},
      issn = {0550-3213},
      coden = {NUPBBO},
      mrclass = {81E13 (17B70 58G37 81D15)},
      mrnumber = {0757857},
      doi = {10.1016/0550-3213(84)90052-X},
      zblnumber = {0661.17013},
      }
  • [CarEil56] H. Cartan and S. Eilenberg, Homological Algebra, Princeton, NJ: Princeton Univ. Press, 1956, vol. 19.
    @book{CarEil56, mrkey = {0077480},
      author = {Cartan, Henri and Eilenberg, Samuel},
      title = {Homological Algebra},
      SERIES={Princeton Math. Series},
      VOLUME={19},
      publisher = {Princeton Univ. Press},
      ADDRESS={Princeton, NJ},
      year = {1956},
      pages = {xv+390},
      mrclass = {09.0X},
      mrnumber = {0077480},
      mrreviewer = {G. Hochschild},
      zblnumber = {0075.24305},
      }
  • [DAnDe-De-07] A. D’Andrea, C. De Concini, A. De Sole, R. Heluani, and V. Kac, Three equivalent definitions of finite W-algebras, 2006.
    @misc{DAnDe-De-07,
      author = {D'Andrea, Alessandro and De Concini, Corrado and De Sole, Alberto and Heluani, Reimundo and Kac, Victor~G.},
      title = {Three equivalent definitions of finite {W}-algebras},
      note = {appendix to \cite{De-Kac06}},
      year = {2006},
      }
  • [DonLamTan04] Go to document C. Dong, C. H. Lam, K. Tanabe, H. Yamada, and K. Yokoyama, "$\Bbb Z_3$ symmetry and $W_3$ algebra in lattice vertex operator algebras," Pacific J. Math., vol. 215, iss. 2, pp. 245-296, 2004.
    @article{DonLamTan04, mrkey = {2068782},
      author = {Dong, Chongying and Lam, Ching Hung and Tanabe, Kenichiro and Yamada, Hiromichi and Yokoyama, Kazuhiro},
      title = {{$\Bbb Z\sb 3$} symmetry and {$W\sb 3$} algebra in lattice vertex operator algebras},
      journal = {Pacific J. Math.},
      fjournal = {Pacific Journal of Mathematics},
      volume = {215},
      year = {2004},
      number = {2},
      pages = {245--296},
      issn = {0030-8730},
      coden = {PJMAAI},
      mrclass = {17B69},
      mrnumber = {2068782},
      mrreviewer = {Julius B. Borcea},
      doi = {10.2140/pjm.2004.215.245},
      zblnumber = {1055.17013},
      }
  • [De-Kac06] Go to document A. De Sole and V. G. Kac, "Finite vs affine $W$-algebras," Jpn. J. Math., vol. 1, iss. 1, pp. 137-261, 2006.
    @article{De-Kac06, mrkey = {2261064},
      author = {De Sole, Alberto and Kac, Victor G.},
      title = {Finite vs affine {$W$}-algebras},
      journal = {Jpn. J. Math.},
      fjournal = {Japanese Journal of Mathematics},
      volume = {1},
      year = {2006},
      number = {1},
      pages = {137--261},
      issn = {0289-2316},
      mrclass = {17B69 (17B68)},
      mrnumber = {2261064},
      mrreviewer = {Hai Sheng Li},
      doi = {10.1007/s11537-006-0505-2},
      zblnumber = {1161.17015},
      }
  • [Duf77] Go to document M. Duflo, "Sur la classification des idéaux primitifs dans l’algèbre enveloppante d’une algèbre de Lie semi-simple," Ann. of Math., vol. 105, iss. 1, pp. 107-120, 1977.
    @article{Duf77, mrkey = {0430005},
      author = {Duflo, Michel},
      title = {Sur la classification des idéaux primitifs dans l'algèbre enveloppante d'une algèbre de {L}ie semi-simple},
      journal = {Ann. of Math.},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {105},
      year = {1977},
      number = {1},
      pages = {107--120},
      issn = {0003-486X},
      mrclass = {17B35},
      mrnumber = {0430005},
      mrreviewer = {N. N. Sapovalov},
      doi = {10.2307/1971027},
      zblnumber = {0346.17011},
      }
  • [FF90] Go to document B. Feigin and E. Frenkel, "Quantization of the Drinfel\cprime d-Sokolov reduction," Phys. Lett. B, vol. 246, iss. 1-2, pp. 75-81, 1990.
    @article{FF90, mrkey = {1071340},
      author = {Feigin, Boris and Frenkel, Edward},
      title = {Quantization of the {D}rinfel\cprime d-{S}okolov reduction},
      journal = {Phys. Lett. B},
      fjournal = {Physics Letters. B},
      volume = {246},
      year = {1990},
      number = {1-2},
      pages = {75--81},
      issn = {0370-2693},
      coden = {PYLBAJ},
      mrclass = {17B65 (17B81 58F07 81R10 81T70)},
      mrnumber = {1071340},
      mrreviewer = {Alice Fialowski},
      doi = {10.1016/0370-2693(90)91310-8},
      zblnumber = {1242.17023},
      }
  • [FKW92] Go to document E. Frenkel, V. Kac, and M. Wakimoto, "Characters and fusion rules for $W$-algebras via quantized Drinfel\cprime d-Sokolov reduction," Comm. Math. Phys., vol. 147, iss. 2, pp. 295-328, 1992.
    @article{FKW92, mrkey = {1174415},
      author = {Frenkel, Edward and Kac, Victor and Wakimoto, Minoru},
      title = {Characters and fusion rules for {$W$}-algebras via quantized {D}rinfel\cprime d-{S}okolov reduction},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {147},
      year = {1992},
      number = {2},
      pages = {295--328},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {17B68 (58F07 81R10)},
      mrnumber = {1174415},
      mrreviewer = {D. V. Yur{\cprime}ev},
      url = {http://projecteuclid.org/euclid.cmp/1104250638},
      zblnumber = {0768.17008},
      }
  • [FatLyk88] Go to document V. A. Fateev and S. L. Lykyanov, "The models of two-dimensional conformal quantum field theory with $Z_n$ symmetry," Internat. J. Modern Phys. A, vol. 3, iss. 2, pp. 507-520, 1988.
    @article{FatLyk88, mrkey = {0932661},
      author = {Fateev, V. A. and Lykyanov, S. L.},
      title = {The models of two-dimensional conformal quantum field theory with {$Z\sb n$} symmetry},
      journal = {Internat. J. Modern Phys. A},
      fjournal = {International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. Nuclear Physics},
      volume = {3},
      year = {1988},
      number = {2},
      pages = {507--520},
      issn = {0217-751X},
      mrclass = {81E99 (81D15 81E40 82A25 82A69)},
      mrnumber = {0932661},
      mrreviewer = {Philippe Christe},
      doi = {10.1142/S0217751X88000205},
      }
  • [FreZhu92] Go to document I. B. Frenkel and Y. Zhu, "Vertex operator algebras associated to representations of affine and Virasoro algebras," Duke Math. J., vol. 66, iss. 1, pp. 123-168, 1992.
    @article{FreZhu92, mrkey = {1159433},
      author = {Frenkel, Igor B. and Zhu, Yongchang},
      title = {Vertex operator algebras associated to representations of affine and {V}irasoro algebras},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {66},
      year = {1992},
      number = {1},
      pages = {123--168},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {17B68 (17B67)},
      mrnumber = {1159433},
      mrreviewer = {Geoffrey Mason},
      doi = {10.1215/S0012-7094-92-06604-X},
      zblnumber = {0848.17032},
      }
  • [GanGin02] Go to document W. L. Gan and V. Ginzburg, "Quantization of Slodowy slices," Int. Math. Res. Not., vol. 2002, iss. 5, pp. 243-255, 2002.
    @article{GanGin02, mrkey = {1876934},
      author = {Gan, Wee Liang and Ginzburg, Victor},
      title = {Quantization of {S}lodowy slices},
      journal = {Int. Math. Res. Not.},
      fjournal = {International Mathematics Research Notices},
      year = {2002},
      number = {5},
      pages = {243--255},
      issn = {1073-7928},
      mrclass = {53D20 (17B35 17B63 53D50)},
      mrnumber = {1876934},
      mrreviewer = {William M. McGovern},
      doi = {10.1155/S107379280210609X},
      volume = {2002},
      zblnumber = {0989.17014},
      }
  • [Gin08] Go to document V. Ginzburg, "Harish-Chandra bimodules for quantized Slodowy slices," Represent. Theory, vol. 13, pp. 236-271, 2009.
    @article{Gin08, mrkey = {2515934},
      author = {Ginzburg, Victor},
      title = {Harish-{C}handra bimodules for quantized {S}lodowy slices},
      journal = {Represent. Theory},
      fjournal = {Representation Theory. An Electronic Journal of the American Mathematical Society},
      volume = {13},
      year = {2009},
      pages = {236--271},
      issn = {1088-4165},
      mrclass = {17B08 (14A22 17B10)},
      mrnumber = {2515934},
      mrreviewer = {Arvid Siqveland},
      doi = {10.1090/S1088-4165-09-00355-0},
      zblnumber = {1250.17007},
      }
  • [GorKac0905] Go to document M. Gorelik and V. Kac, "On complete reducibility for infinite-dimensional Lie algebras," Adv. Math., vol. 226, iss. 2, pp. 1911-1972, 2011.
    @article{GorKac0905, mrkey = {2737804},
      author = {Gorelik, Maria and Kac, Victor},
      title = {On complete reducibility for infinite-dimensional {L}ie algebras},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {226},
      year = {2011},
      number = {2},
      pages = {1911--1972},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {17B65 (17B10 17B69)},
      mrnumber = {2737804},
      mrreviewer = {David George Taylor},
      doi = {10.1016/j.aim.2010.09.001},
      zblnumber = {1225.17026},
      }
  • [Kos78] Go to document B. Kostant, "On Whittaker vectors and representation theory," Invent. Math., vol. 48, iss. 2, pp. 101-184, 1978.
    @article{Kos78, mrkey = {0507800},
      author = {Kostant, Bertram},
      title = {On {W}hittaker vectors and representation theory},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {48},
      year = {1978},
      number = {2},
      pages = {101--184},
      issn = {0020-9910},
      coden = {INVMBH},
      mrclass = {22E47 (22E45)},
      mrnumber = {0507800},
      mrreviewer = {A. U. Klimyk},
      doi = {10.1007/BF01390249},
      zblnumber = {0405.22013},
      }
  • [KacRoaWak03] Go to document V. Kac, S. Roan, and M. Wakimoto, "Quantum reduction for affine superalgebras," Comm. Math. Phys., vol. 241, iss. 2-3, pp. 307-342, 2003.
    @article{KacRoaWak03, mrkey = {2013802},
      author = {Kac, Victor and Roan, Shi-Shyr and Wakimoto, Minoru},
      title = {Quantum reduction for affine superalgebras},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {241},
      year = {2003},
      number = {2-3},
      pages = {307--342},
      issn = {0010-3616},
      coden = {CMPHAY},
      mrclass = {17B65 (17B37 17B67 81R10 81T70)},
      mrnumber = {2013802},
      mrreviewer = {David Hernandez},
      zblnumber = {1106.17026},
      doi = {10.1007/s00220-003-0926-1},
     }
  • [KosSte87] Go to document B. Kostant and S. Sternberg, "Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras," Ann. Physics, vol. 176, iss. 1, pp. 49-113, 1987.
    @article{KosSte87, mrkey = {0893479},
      author = {Kostant, Bertram and Sternberg, Shlomo},
      title = {Symplectic reduction, {BRS} cohomology, and infinite-dimensional {C}lifford algebras},
      journal = {Ann. Physics},
      fjournal = {Annals of Physics},
      volume = {176},
      year = {1987},
      number = {1},
      pages = {49--113},
      issn = {0003-4916},
      coden = {ADNYA6},
      mrclass = {58F05 (15A66 17B56 58F06 81D07 81E99)},
      mrnumber = {0893479},
      mrreviewer = {Andrew Pressley},
      doi = {10.1016/0003-4916(87)90178-3},
      zblnumber = {0642.17003},
      }
  • [KacWak89] V. G. Kac and M. Wakimoto, "Classification of modular invariant representations of affine algebras," in Infinite-Dimensional Lie Algebras and Groups, Teaneck, NJ: World Sci. Publ., 1989, vol. 7, pp. 138-177.
    @incollection{KacWak89, mrkey = {1026952},
      author = {Kac, V. G. and Wakimoto, M.},
      title = {Classification of modular invariant representations of affine algebras},
      booktitle = {Infinite-Dimensional {L}ie Algebras and Groups},
      venue = {{L}uminy-{M}arseille, 1988},
      series = {Adv. Ser. Math. Phys.},
      volume = {7},
      pages = {138--177},
      publisher = {World Sci. Publ.},
      address = {Teaneck, NJ},
      year = {1989},
      mrclass = {17B67 (11F22)},
      mrnumber = {1026952},
      mrreviewer = {Vyjayanthi Chari},
      zblnumber = {0742.17022},
      }
  • [KacWak08] Go to document V. G. Kac and M. Wakimoto, "On rationality of $W$-algebras," Transform. Groups, vol. 13, iss. 3-4, pp. 671-713, 2008.
    @article{KacWak08, mrkey = {2452611},
      author = {Kac, Victor G. and Wakimoto, Minoru},
      title = {On rationality of {$W$}-algebras},
      journal = {Transform. Groups},
      fjournal = {Transformation Groups},
      volume = {13},
      year = {2008},
      number = {3-4},
      pages = {671--713},
      issn = {1083-4362},
      mrclass = {17B69 (11F22)},
      mrnumber = {2452611},
      mrreviewer = {Pierluigi M{ö}seneder Frajria},
      doi = {10.1007/s00031-008-9028-7},
      zblnumber = {1221.17033},
      }
  • [LukFat89] S. L. Lukcprimeyanov and V. A. Fateev, "Exactly soluble models of conformal quantum field theory associated with the simple Lie algebra $D_n$," Yadernaya Fiz., vol. 49, iss. 5, pp. 1491-1504, 1989.
    @article{LukFat89, mrkey = {1020036},
      author = {Luk{\cprime}yanov, S. L. and Fateev, V. A.},
      title = {Exactly soluble models of conformal quantum field theory associated with the simple {L}ie algebra {$D\sb n$}},
      journal = {Yadernaya Fiz.},
      fjournal = {Akademiya Nauk SSSR. Yadernaya Fizika},
      volume = {49},
      year = {1989},
      number = {5},
      pages = {1491--1504},
      coden = {IDFZA7},
      mrclass = {81T40 (82B20 82B23)},
      mrnumber = {1020036},
      mrreviewer = {Raiko Zaikov},
      }
  • [Li97] Go to document H. Li, "The physics superselection principle in vertex operator algebra theory," J. Algebra, vol. 196, iss. 2, pp. 436-457, 1997.
    @article{Li97, mrkey = {1475118},
      author = {Li, Haisheng},
      title = {The physics superselection principle in vertex operator algebra theory},
      journal = {J. Algebra},
      fjournal = {Journal of Algebra},
      volume = {196},
      year = {1997},
      number = {2},
      pages = {436--457},
      issn = {0021-8693},
      coden = {JALGA4},
      mrclass = {17B69 (81R10)},
      mrnumber = {1475118},
      mrreviewer = {Mirko Primc},
      doi = {10.1006/jabr.1997.7126},
      zblnumber = {0885.17019},
      }
  • [Los11] Go to document I. Losev, "Finite-dimensional representations of $W$-algebras," Duke Math. J., vol. 159, iss. 1, pp. 99-143, 2011.
    @article{Los11, mrkey = {2817650},
      author = {Losev, Ivan},
      title = {Finite-dimensional representations of {$W$}-algebras},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {159},
      year = {2011},
      number = {1},
      pages = {99--143},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {17B35 (17B37)},
      mrnumber = {2817650},
      mrreviewer = {Simon M. Goodwin},
      doi = {10.1215/00127094-1384800},
      ZBLNUMBER = {1235.17007},
     }
  • [MatNagTsu05] A. Matsuo, K. Nagatomo, and A. Tsuchiya, "Quasi-finite algebras graded by Hamiltonian and vertex operator algebras," in Moonshine: The First Quarter Century and Beyond, Cambridge: Cambridge Univ. Press, 2010, vol. 372, pp. 282-329.
    @incollection{MatNagTsu05, mrkey = {2681785},
      author = {Matsuo, Atsushi and Nagatomo, Kiyokazu and Tsuchiya, Akihiro},
      title = {Quasi-finite algebras graded by {H}amiltonian and vertex operator algebras},
      booktitle = {Moonshine: The First Quarter Century and Beyond},
      series = {London Math. Soc. Lecture Note Ser.},
      volume = {372},
      pages = {282--329},
      publisher = {Cambridge Univ. Press},
      address = {Cambridge},
      year = {2010},
      mrclass = {17B69 (16D90)},
      mrnumber = {2681785},
      mrreviewer = {William J. Cook},
      zblnumber = {1227.17013},
      }
  • [NagTsu05] Go to document K. Nagatomo and A. Tsuchiya, "Conformal field theories associated to regular chiral vertex operator algebras. I. Theories over the projective line," Duke Math. J., vol. 128, iss. 3, pp. 393-471, 2005.
    @article{NagTsu05, mrkey = {2145740},
      author = {Nagatomo, Kiyokazu and Tsuchiya, Akihiro},
      title = {Conformal field theories associated to regular chiral vertex operator algebras. {I}. {T}heories over the projective line},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {128},
      year = {2005},
      number = {3},
      pages = {393--471},
      issn = {0012-7094},
      coden = {DUMJAO},
      mrclass = {81T40 (17B69 81R10)},
      mrnumber = {2145740},
      mrreviewer = {Hai Sheng Li},
      doi = {10.1215/S0012-7094-04-12831-3},
      zblnumber = {1074.81065},
      }
  • [Pre02] Go to document A. Premet, "Special transverse slices and their enveloping algebras," Adv. Math., vol. 170, iss. 1, pp. 1-55, 2002.
    @article{Pre02, mrkey = {1929302},
      author = {Premet, Alexander},
      title = {Special transverse slices and their enveloping algebras},
      note = {with an appendix by Serge Skryabin},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {170},
      year = {2002},
      number = {1},
      pages = {1--55},
      issn = {0001-8708},
      coden = {ADMTA4},
      mrclass = {17B35 (16S80)},
      mrnumber = {1929302},
      mrreviewer = {William M. McGovern},
      doi = {10.1006/aima.2001.2063},
      zblnumber = {1005.17007},
      }
  • [Skr02] S. Skryabin, A category equivalence, 2002.
    @misc{Skr02,
      author = {Skryabin, Serge},
      note = {appendix to \cite{Pre02}},
      year = {2002},
      title = {A category equivalence},
      }
  • [Wan93] Go to document W. Wang, "Rationality of Virasoro vertex operator algebras," Internat. Math. Res. Notices, iss. 7, pp. 197-211, 1993.
    @article{Wan93, mrkey = {1230296},
      author = {Wang, Weiqiang},
      title = {Rationality of {V}irasoro vertex operator algebras},
      journal = {Internat. Math. Res. Notices},
      fjournal = {International Mathematics Research Notices},
      year = {1993},
      number = {7},
      pages = {197--211},
      issn = {1073-7928},
      mrclass = {17B68 (81R10)},
      mrnumber = {1230296},
      mrreviewer = {Chong Ying Dong},
      doi = {10.1155/S1073792893000212},
      ZBLNUMBER = {0791.17029},
     }
  • [Zhu96] Go to document Y. Zhu, "Modular invariance of characters of vertex operator algebras," J. Amer. Math. Soc., vol. 9, iss. 1, pp. 237-302, 1996.
    @article{Zhu96, mrkey = {1317233},
      author = {Zhu, Yongchang},
      title = {Modular invariance of characters of vertex operator algebras},
      journal = {J. Amer. Math. Soc.},
      fjournal = {Journal of the American Mathematical Society},
      volume = {9},
      year = {1996},
      number = {1},
      pages = {237--302},
      issn = {0894-0347},
      mrclass = {17B69 (11F22 17B68)},
      mrnumber = {1317233},
      mrreviewer = {Chong Ying Dong},
      doi = {10.1090/S0894-0347-96-00182-8},
      zblnumber = {0854.17034},
      }

Authors

Tomoyuki Arakawa

Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan