A comprehensive research of vertex algebras, especially the W-algebras

2012 Fiscal Year Final Research Report

• Project/Area Number: 20340007
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto University (2010-2012); Nara Women's University (2008-2009)
• Principal Investigator: ARAKAWA Tomoyuki 京都大学, 数理解析研究所, 准教授 (40377974)
• Co-Investigator(Kenkyū-buntansha): MATSUO Atsushi 東京大学, 数理科学研究科, 准教授 (20238968) ; SUZUKI Takeshi 岡山大学, 自然科学研究科, 准教授 (30335294) ; YAMAUCHI Hiroshi 東京女子大学, 現代教養部, 講師 (40452213)
• Co-Investigator(Renkei-kenkyūsha): YAMADA Hiromichi 一橋大学, 経済学研究科, 教授 (50134888) ; MIYAMOTO Masahiko 筑波大学, 数理物質科学研究科, 教授 (30125356) ; MATUZAWA Jyunichi 奈良女子大学, 理学部, 教授 (00212217) ; KONNO Hitoshi 広島大学, 理学研究科, 准教授 (00291477)
• Project Period (FY): 2008 – 2012
• Keywords: W 代数 / アフィンリー環 / 頂点代数

Research Abstract

On admissible representations of affine Kac-Moody algebras, we proved the he conjecture of Frenkel, Kac and Wakimoto on the existence of two-sided BGG resolutions, the conjecture of Adamovic and Milas on the corresponding vertex operators algebra, and the conjecture of Feigin and Frenkel on their singular supports. On critical level representations of affine Kac-Moody algebras, we proved the new linkage principal and established a chiral Borel-Weil-Bott theorem. On W-algebras, we prove the C_2-cofinitness of the exceptional W-algebras discovered by Kac and Wakimoto and prove the admissible representations of affine Kac-Moody algebras. We obtained various results on the W-algebras at the critical levels.

Research Products

• [Journal Article] On the restricted Verma modules at the critical level (2012)
  • Author(s): T. Arakawa, P. Fiebig
  • Journal Title: Trans. Amer. Math. Soc. Volume: 364 Pages: 4683-4712
  • DOI: DOI:10.1090/S0002-9947-2012-05467-5
  • Peer Reviewed
• [Journal Article] A chiral Borel-Weil-Bott theorem (2012)
  • Author(s): T. Arakawa, F. Malikov
  • Journal Title: Adv. Math. Volume: 229 Pages: 2908-2949
  • DOI: DOI:10.1016/j.aim.2011.11.002
  • Peer Reviewed
• [Journal Article] The linkage principle for restricted critical level representations of affine Kac-Moody algebras (2012)
  • Author(s): T. Arakawa, P. Fiebig
  • Journal Title: Compos. Math. Volume: 148 Pages: 1787-1810
  • DOI: DOI:10.1112/S0010437X12000395
  • Peer Reviewed
• [Journal Article] A remark on the $C_2$-cofiniteness condition on vertex algebras (2012)
  • Author(s): Tomoyuki Arakawa
  • Journal Title: Math. Z. Volume: vol. 270, no. 1-2 Pages: 559-575
  • DOI: DOI:10.1007/s00209-010-0812-4
  • Peer Reviewed
• [Journal Article] W-algebras at the critical level (2012)
  • Author(s): Tomoyuki Arakawa
  • Journal Title: Contemp. Math. Volume: 565 Pages: 1-14
  • URL: http://arxiv.org/pdf/1111.6329v1.pdf
  • Peer Reviewed
• [Journal Article] McKay's E7 observation on the Babymonster (2012)
  • Author(s): Gerald Hoehn, Ching Hung Lam and Hiroshi Yamauchi
  • Journal Title: International Mathematics Research Notices Volume: Vol. 2012 Pages: 166-212
  • DOI: DOI:10.1093/imrn/rnr009
  • Peer Reviewed
• [Journal Article] McKay's E6 observation on the largest Fischer group (2012)
  • Author(s): Gerald Hoehn, Ching Hung Lam and Hiroshi Yamauchi
  • Journal Title: Communications in Mathematical Physics Volume: Vol. 2 Pages: 329-365
  • DOI: DOI:10.1007/s00220-011-1413-8
  • Peer Reviewed
• [Journal Article] Representation theory of W-algebras, II (2011)
  • Author(s): Tomoyuki Arakawa
  • Journal Title: Adv. Stud. Pure Math. Volume: 61 Pages: 51-90
  • Peer Reviewed
• [Journal Article] Algebras of twisted chiral differential operators and affine localization of $g$-modules (2011)
  • Author(s): T. Arakawa, D. Chebotarov and F. Malikov
  • Journal Title: Sel. Math. New Ser. Volume: vol.17, no. 1 Pages: 1-46
  • DOI: DOI:10.1007/s00029-010-0040-0
  • Peer Reviewed
• [Journal Article] The FLM conjecture and framed VOA, Vertex operator algebras and related areas (2009)
  • Author(s): Ching Hung Lam and Hiroshi Yamauchi
  • Journal Title: Contemporary Mathematics,Amer. Math. Soc., Providence, RI Pages: 125-138
  • URL: http://www.math.twcu.ac.jp/~yamauchi/math/illinois2008/illinois2008.pdf
  • Peer Reviewed
• [Journal Article] On the structure of framed vertex operator algebras and their pointwise frame stabilizers (2008)
  • Author(s): Ching Hung Lam and Hiroshi Yamauchi
  • Journal Title: Communications in Mathematical Physics Volume: 277 Pages: 237-285
  • DOI: DOI:10.1007/s00220-007-0323-2
  • Peer Reviewed
• [Journal Article] Binary codes and the classification of holomorphic framed vertex operator algebras (2008)
  • Author(s): Ching Hung Lam and Hiroshi Yamauchi
  • Journal Title: RIMS Kokyoroku Bessatsu B8 Pages: 131-150
  • URL: http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B8/pdf/B8-11.pdf
  • Peer Reviewed
• [Journal Article] Rationality of Bershadsky-Polyakov vertex algebras
  • Author(s): Tomoyuki Arakawa
  • Journal Title: to appear in Comm. Math. Phys.
  • URL: http://arxiv.org/pdf/1005.0185.pdf
  • Peer Reviewed
• [Journal Article] Combinatorics for Graded Cartan Matrices of the Iwahori-Hecke Algebra of Type A
  • Author(s): M.Ando, T.Suzuki, H-F.Yamada
  • Journal Title: to appear in Annals of Combinatorics
  • URL: http://arxiv.org/pdf/1005.1134.pdf
  • Peer Reviewed
• [Presentation] On 3-transposition groups based on the 3A-algebra (2013)
  • Author(s): Hiroshi Yamauchi
  • Organizer: Taitung Workshop
  • Place of Presentation: Taitung University, Taitung, Taiwan
  • Year and Date: 2013-03-27
• [Presentation] 代数学賞受賞特別講演「W 代数の表現論」 (2013)
  • Author(s): 荒川 知幸
  • Organizer: 日本数学会年会
  • Place of Presentation: 京都大学
  • Year and Date: 2013-03-22
• [Presentation] Binary codes and classification of framed vertex operator algebras (2013)
  • Author(s): Hiroshi Yamauchi
  • Organizer: Exceptional structures in geometry and field theory, IPMU
  • Year and Date: 2013-03-07
• [Presentation] Catgorie O: geomtrie et catgorification (2012)
  • Author(s): Tomoyuki Arakawa
  • Organizer: A mini-course lecture
  • Place of Presentation: CIRM, Luminy. (France)
  • Year and Date: 20121210-14
• [Presentation] Nilpotent orbits and the Kac-Wakimoto conjecture (2012)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Conference on Groups, VOAs and Related Structures in Honor of Masahiko Miyamoto
  • Place of Presentation: University of Tsukuba, Japan.
  • Year and Date: 20121210-14
• [Presentation] Algebraic Structures in Integrable Systems (2012)
  • Author(s): Tomoyuki Arakawa
  • Organizer: A mini-course lecture
  • Place of Presentation: Moscow (Russia)
  • Year and Date: 20121203-07
• [Presentation] Rationality of W -algebras (2012)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Conference on Vertex Operator Algebra and Related Topics
  • Place of Presentation: Shanghai Jiao Tong University, China.
  • Year and Date: 20120824-27
• [Presentation] Rationality of W -algebras (2012)
  • Author(s): Tomoyuki Arakawa
  • Organizer: The XXIX International Colloquium on Group-Theoretical Methods in Physics
  • Place of Presentation: Chern Institute of Mathematics Tianjin, China
  • Year and Date: 20120820-26
• [Presentation] Localization of W-algebras at the critical level (2012)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Finite Groups and Vertex Operator Algebras in Honor of Professor Hiromichi Yamada's 60th birthday
  • Place of Presentation: Tokyo Woman's Christian University, Tokyo, Japan
  • Year and Date: 20120217-18
• [Presentation] VOAs and Related Structures in Honor of Masahiko Miyamoto (2012)
  • Author(s): Hiroshi Yamauchi
  • Organizer: Miyamoto involutions,Conference on Groups
  • Place of Presentation: 筑波大学
  • Year and Date: 2012-09-10
• [Presentation] 3A-algebra and 3-transposition property (2012)
  • Author(s): Hiroshi Yamauchi
  • Organizer: Conference on Vertex Operator Algebras and Related Topics
  • Place of Presentation: Shanghai Jiao Tong University, Shanghai, China
  • Year and Date: 2012-08-25
• [Presentation] Extended Griess algebras, conformal designs and Matsuo-Norton trace formulae (2012)
  • Author(s): Hiroshi Yamauchi
  • Organizer: The XXIX International Colloquium on GroupTheoretical Methods in Physics, Chern Institute of Mathematics
  • Place of Presentation: Nankai University, Tianjin, China
  • Year and Date: 2012-08-20
• [Presentation] 共形デザインについて (2012)
  • Author(s): 山内 博
  • Organizer: 第24回有限群論草津セミナー
  • Place of Presentation: 草津セミナーハウス
  • Year and Date: 2012-07-28
• [Presentation] 拡大グライス代数と松尾-ノートンの跡公式 (2012)
  • Author(s): 山内 博
  • Organizer: 有限群とその表現、頂点作用素代数、組合せ論の研究集会
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2012-03-05
• [Presentation] 3A 代数と 3 互換群有限群と頂点作用素代数 (2012)
  • Author(s): 山内 博
  • Place of Presentation: 東京女子大学
  • Year and Date: 2012-02-17
• [Presentation] Rationality of admissible affine vertex algebras in the category O (2011)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Conference on Vertex Operator Algebras, Finite Groups and Related Topics
  • Place of Presentation: Academia Sinica, Taiwan
  • Year and Date: 20111218-22
• [Presentation] Matsuo-Norton trace formula for vertex operator superalgebras, Conference on Vertex Operator Algebras, Finite Groups and Related Topics (2011)
  • Author(s): H. Yamauchi
  • Organizer: Institute of Mathematics
  • Place of Presentation: Academia Sinica, Taipei, Taiwan
  • Year and Date: 2011-12-19
• [Presentation] 分割の数え上げと対称群のカルタン行列 (2011)
  • Author(s): 鈴木武史
  • Organizer: 信州代数セミナー
  • Place of Presentation: 信州大学
  • Year and Date: 2011-11-14
• [Presentation] Affine W-algebras (2011)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Algebra Seminar
  • Place of Presentation: The University of York, UK
  • Year and Date: 2011-10-24
• [Presentation] Affine W-algebras (2011)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Algebra Seminar
  • Place of Presentation: Lancaster University, UK.
  • Year and Date: 2011-10-21
• [Presentation] Affine W-algebras (2011)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Algebra Seminar
  • Place of Presentation: The University of Edinburgh, UK.
  • Year and Date: 2011-10-18
• [Presentation] Affine W-algebras (2011)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Algebra Seminar
  • Place of Presentation: Birmingham University, UK.
  • Year and Date: 2011-10-13
• [Presentation] Affine W-algebras (2011)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Algebra Seminar
  • Place of Presentation: The University of Manchester, UK.
  • Year and Date: 2011-10-11
• [Presentation] カイラルハミルトニアン構成法と表現論-W 代数をめぐって (2011)
  • Author(s): 荒川 知幸
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-10-01
• [Presentation] Trace identity and axial vectors for the Baby-monster, Conformal Field Theory, Automorphic Forms and Related Topics (2011)
  • Author(s): H. Yamauchi
  • Organizer: The MAThematics Center Heidelberg
  • Place of Presentation: Heidelberg University, German
  • Year and Date: 2011-09-19
• [Presentation] Localization of affine W-algebras at the critical level (2011)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Perspectives in Algebraic Lie Theory, Isaac Newton Institute for Mathematical Sciences
  • Place of Presentation: UK
  • Year and Date: 2011-09-15
• [Presentation] Localization of affine W-algebras at the critical level (2011)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Algebraic Geometry and Representation Theory Seminar
  • Place of Presentation: Weizmann Institute of Science, Israel
  • Year and Date: 2011-05-27
• [Presentation] Admissible affine vertex algebras of type A (2011)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Algebraic Geometry and Representation TheorySeminar
  • Place of Presentation: Weizmann Institute of Science, Israel
  • Year and Date: 2011-05-20
• [Presentation] カイラルハミルトニアン還元法と表現論-W 代数をめぐって- (2011)
  • Author(s): 荒川 知幸
  • Organizer: 談話会, 名古屋大学大学院多元数理科学研究科
  • Year and Date: 2011-05-11
• [Presentation] Rationality and smoothness of W-algebras (2010)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Representation Theory of Algebraic Groups and Quantum Groups
  • Place of Presentation: Nagoya, Japan
  • Year and Date: 20100802-06
• [Presentation] Kac-Wakimoto conjecture and associated varieties of modules over KacMoody algebras (2010)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Research into Vertex Operator Algebras, Finite Groups and Combinatorics
  • Place of Presentation: Kyoto
  • Year and Date: 2010-12-16
• [Presentation] BRST reduction and representation theory of W-algebras (2010)
  • Author(s): 荒川 知幸
  • Organizer: 談話会, RIMS
  • Place of Presentation: 京都大学
  • Year and Date: 2010-10-13
• [Presentation] 有限群と軌道体理論 (2010)
  • Author(s): 山内 博
  • Organizer: 第22回有限群論草津セミナー
  • Place of Presentation: 草津セミナーハウス
  • Year and Date: 2010-07-30
• [Presentation] Associated varieties of Kac-Wakimoto admissible representations (2010)
  • Author(s): Tomoyuki Arakawa
  • Organizer: USC Algebra Seminar
  • Place of Presentation: University of Southern California, USA.
  • Year and Date: 2010-04-19
• [Presentation] 散在型有限単純群と頂点作用素代数 (2010)
  • Author(s): 山内博
  • Organizer: 日本数学会 2010 年度年会
  • Place of Presentation: 慶應義塾大学
  • Year and Date: 2010-03-27
• [Presentation] Nilpotent orbits and Kac-Wakimoto admissible representations of Kac-Moody algebras (2010)
  • Author(s): Tomoyuki Arakawa
  • Organizer: 第9回巾零軌道と表現論研究集会
  • Place of Presentation: 北海道大学
  • Year and Date: 2010-02-19
• [Presentation] Lie, Hecke and Cherednik Algebras in Conformal Field Theory (2009)
  • Author(s): T.Suzuki
  • Organizer: Tokyo-Seoul Confenece in Mathematics, Representation Theory
  • Place of Presentation: 東京大学
  • Year and Date: 2009-12-15
• [Presentation] n-transposition groups and vertex operator algebras, Lie algebras, vertex algebras and automorphic forms (2009)
  • Author(s): H. Yamauchi
  • Organizer: 15 South College Street, Edinburgh
  • Place of Presentation: UK
  • Year and Date: 2009-09-04
• [Presentation] Admissible representation of affine VOAs and modular invariance,Algebras (2009)
  • Author(s): H. Yamauchi
  • Organizer: Groups and Geometries 2009
  • Place of Presentation: 玉原国際セミナーハウス
  • Year and Date: 2009-08-20
• [Presentation] Varieties of nilpotent orbits, Kac-Moody algebra and C2-cofiniteness of W-algebras (2009)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Mini Conference on chiral differential operators, IPMU
  • Year and Date: 2009-08-18
• [Presentation] 旗多様体上のカイラル微分作用とアフィン Lie 環の表現論 (2009)
  • Author(s): 荒川 知幸
  • Organizer: 代数学シンポジウム
  • Place of Presentation: 明治大学
  • Year and Date: 2009-08-06
• [Presentation] 互換群と巾等元 (2009)
  • Author(s): 山内博
  • Organizer: 第21回有限群論草津セミナー
  • Place of Presentation: 草津セミナーハウス
  • Year and Date: 2009-08-02
• [Presentation] Varieties of nilpotent orbits, Kac-Moody algebras, and W-algebras (2009)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Workshop and Summer School on Lie Theory and Representation Theory II
  • Place of Presentation: East China Normal University, Shanghai, China
  • Year and Date: 2009-07-20
• [Presentation] Affine W-algebras and Zhu's Poisson varieties associated with Kac- Moody vertex algebras (2009)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Algebraic Lie Structures with Origins in Physics
  • Place of Presentation: Isaac Newton Institute for Mathematical Sciences, UK.
  • Year and Date: 2009-03-23
• [Presentation] Chiral differential operators and affine chiralization of g-modules (2008)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Taipei Workshop in Lie Theory
  • Place of Presentation: Institute of Mathematics, Academia Sinica, Taipei, Taiwan.
  • Year and Date: 2008-12-28
• [Presentation] W 代数の指標公式について (2008)
  • Author(s): 荒川 知幸
  • Organizer: 談話会
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2008-12-03
• [Presentation] W 代数の表現論について (2008)
  • Author(s): 荒川 知幸
  • Organizer: 2008 年数学会秋季総合 分科会代数学分科会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-26
• [Presentation] On the transformation property of the Lie algebras associated with vertex operator algebra (2008)
  • Author(s): A. Matuso
  • Organizer: International Conference on Vertex Operator Algebras and Related Areas
  • Place of Presentation: イリノイ州立大学
  • Year and Date: 2008-07-10
• [Presentation] Representations of W-algebras and conjectures of Kac-Wakimoto (2008)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Conference "Symmetries in Mathematics and Physics" to honor the 65th birthday of Prof. Victor G. Kac
  • Place of Presentation: Cortona, Italy
  • Year and Date: 2008-06-24
• [Presentation] Representations of W-algebras and affine Kac-Moody algebras (2008)
  • Author(s): Tomoyuki Arakawa
  • Organizer: Erlangen Representation Theory Days
  • Place of Presentation: Emmy-Noether-Zentrum, Universitat Erlangen- Nurnberg, Germany.
  • Year and Date: 2008-06-13
• [Remarks]
  • URL: http://www.kurims.kyoto-u.ac.jp/~arakawa/