Representations of quantum groups and quantum integrable systems

• Project/Area Number: 19540021
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Nagoya University
• Principal Investigator: NAKANISHI Tomoki Nagoya University, 多元数理科学研究科, 准教授 (80227842)
• Project Period (FY): 2007 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 量子群 / 量子可積分系 / 団代数 / 団圏 / Y-system / T-system / dilogarithm / Poisson構造 / Lotka-Volterra方程式 / 多元環 / クラスター代数 / 指標

Research Abstract

We clarify the relation between cluster algebras and the T- and Y-systems, which are functional equations originated from Bethe ansatz method. Using the cluster algebraic formulation, we prove the periodicities of Y-systems and the associated dilogarithm identities, which have remained as conjectures for twenty years. We also clarify various theorems and properties of cluster algebras.

Research Products

• [Journal Article] T-systems, Y-systems, and cluster algebras : Tamely laced case (2011)
  • Author(s): T..Nakanishi
  • Journal Title: In Proc.Of the Infinite Analysis 09, New Trends in Quantum Integrable Systems(eds.B.Feigin, et al.)(World Scientifi, Singapore) Pages: 325-355
  • Peer Reviewed
• [Journal Article] T-systems and Y-systems in integrable systems (2011)
  • Author(s): Atsuo Kuniba, Tomoki Nakanishi, Junji Suzuki
  • Journal Title: Journat of Physics A : Math.Theor. Volume: 44 Pages: 103001-103001
  • Peer Reviewed
• [Journal Article] Periodicities of T-systems and Y-systems (2010)
  • Author(s): R.Inoue, O.Iyama, A.Kuniba, J.Suzuki, T.Nakanishi
  • Journal Title: Naogya Math J. 197 Pages: 56-174
  • Peer Reviewed
• [Journal Article] Dilogarithm identities for sine-Gordon and reduced sine-Gordon Y-systems (2010)
  • Author(s): T..Nakanishi, R.Tateo
  • Journal Title: SIGMA 6 085 Pages: 34-34
  • Peer Reviewed
• [Journal Article] Periodicities in cluster algebras and dilogarithm identities, to appear in Representation of Algebras and Related Topics (2010)
  • Author(s): T..Nakanishi
  • Journal Title: Proc.Of ICRA XIV, Tokyo
  • Peer Reviewed
• [Journal Article] Dilogarithm identities for sine-Gordona dn reduced sine-Gordon Y-systems (2010)
  • Author(s): Tomoki Nakanishi, Roberto Tateo
  • Journal Title: Symmetry, Integrability and Geometry : Methods and Applications Volume: 6 Pages: 85-85
  • Peer Reviewed
• [Journal Article] Periodicities of T-systems and Y-systems (2010)
  • Author(s): R. Inoue, O. Iyama, A. Kuniba, J. Suzuki, T. Nakanishi
  • Journal Title: Naogya Math J. 197 Pages: 56-174
• [Journal Article] Periodicities of T-systems and Y-systems (2010)
  • Author(s): R.Inoue, O.lyama, A.Kuniba, T.Nakanishi, J.Suzuki
  • Journal Title: Nagoya Math J. 197 Pages: 59-174
  • Peer Reviewed
• [Journal Article] T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras (2009)
  • Author(s): A.Kuniba, T..Nakanishi, J.Suzuki
  • Journal Title: SIGMA 5 108 Pages: 23-23
  • Peer Reviewed
• [Journal Article] T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras (2009)
  • Author(s): A. Kuniba, T. Nakanishi , J. Suzuki
  • Journal Title: SIGMA5 108 Pages: 23-23
• [Journal Article] T-systems and Y-systems for quantum affinizations of quantum Kac-Moodyalgebras (2009)
  • Author(s): A.Kuniba, T.Nakanishi, J.Suzuki
  • Journal Title: SIGMA 5(pagesページ番号無) Pages: 108-23
  • Peer Reviewed
• [Journal Article] Paths and tableaux descriptions of Jacobi-Trudi determiant associated with quantum affine algebra of type D_n (2007)
  • Author(s): Wakako Nakai, Tomoki Nakanishi
  • Journal Title: J. Algebr. Comb. 26 Pages: 253-290
  • Peer Reviewed
• [Journal Article] Paths and tableaux descriptions of Jacobi-Trudi determiant associated with quantum affine algebra of type C_n (2007)
  • Author(s): Wakako Nakai, Tomoki Nakanishi
  • Journal Title: SIGMA 3 Pages: 078-20
  • Peer Reviewed
• [Journal Article] On Frenkel-Mukhin algorithm for $q$-character of quantum affine algebras
  • Author(s): W.Nakai, T..Nakanishi
  • Journal Title: to appear in Advanced Studies in Pure Math.
  • Peer Reviewed
• [Journal Article] Dilogarithm identities for conformal field theories and cluster algebras : simply laced case
  • Author(s): T..Nakanishi
  • Journal Title: to appear in Nagoya Math.J.
  • Peer Reviewed
• [Journal Article] Periodicities of T and Y-sysems, dilogarithm identities and cluster algebras I : Type B_r
  • Author(s): R.Inoue, O.Iyama, B.Keller, A.Kuniba, T..Nakanishi
  • Journal Title: to appear in Publ.RIMS.
  • Peer Reviewed
• [Journal Article] Periodicities of T and Y-sysems, dilogarithm identities, and cluster algebras II : Types C_r, F_4, and G_2
  • Author(s): R.Inoue, O.Iyama, B.Keller, A.Kuniba, T..Nakanishi
  • Journal Title: to appear in Publ.RIMS.
  • Peer Reviewed
• [Journal Article] T-systems and Y-systems in integrable systems
  • Author(s): A.Kuniba, T..Nakanishi, J.Suzuki,
  • Journal Title: J.Phys.A : Math. Theor. 44
  • Peer Reviewed
• [Journal Article] Difference equations and cluster algebras I : Piosson bracket for integrable difference equations
  • Author(s): R.Inoue, T..Nakanishi
  • Journal Title: to appear in RIMS Kokyuroku Bessatsu.
  • Peer Reviewed
• [Journal Article] On tropical dualities in cluster algebras
  • Author(s): T..Nakanishi, A.Zelevinsky
  • Journal Title: to appear in Contemp.Math.
  • Peer Reviewed
• [Presentation] Periodicities in cluster algebras and cluster categories from quantum groups (2010)
  • Organizer: Advanced School and Conference on Homological and Geometrical Methods in Representation theory
  • Place of Presentation: トリエステ
• [Presentation] Dilogarithm identities and cluster algebras (2010)
  • Organizer: 日本数学会年会
  • Place of Presentation: 慶応大
• [Presentation] T and Y-systems, dilogarithm identities and cluster algebras : nonsimply laced case (2010)
  • Author(s): Inoue, Iyama, Keller, Kuniba
  • Organizer: 日本数学会年会
  • Place of Presentation: 慶応大
• [Presentation] Dilogarithm identities in conformal field theory and cluster algebras. (2010)
  • Organizer: Quantum groups and quantum topology
  • Place of Presentation: 京大
• [Presentation] Dilogarithm identities in conformal field theory and cluster algebras. (2010)
  • Organizer: Infinite Analysis 10 : Developments in quantum integrable systems
  • Place of Presentation: 京大
• [Presentation] Dilogarithm identities in conformal field theory and cluster algebras. (2010)
  • Organizer: Representation theory of algebraic groups and quantum groups'10
  • Place of Presentation: 名大
• [Presentation] Dilogarithm identities in conformal field theory and cluster algebras. (2010)
  • Organizer: Quantum dilogarithm & quantum Teichmuller theory
  • Place of Presentation: オーフス
• [Presentation] Dilogarithm identities in conformal field theory and cluster algebras. (2010)
  • Organizer: Symposium on Algebra
  • Place of Presentation: 北大
• [Presentation] Periodic cluster algebras and dilogarithm identities. (2010)
  • Organizer: XIV International conference of representations of algebras and workshop
  • Place of Presentation: 東京
• [Presentation] Periodic cluster algebras and dilogarithm identities. (2010)
  • Organizer: 日本数学会
  • Place of Presentation: 名大
• [Presentation] Discrete systems and cluster algebras I.General T-systems and Y-systems. (2010)
  • Organizer: Algebraic and geometric aspects of discrete integrable systems-integrable systems and cluster algebras
  • Place of Presentation: 東大
• [Presentation] Dilogarithm identities in conformal field theory and cluster algebra (2010)
  • Author(s): Tomoki Nakanishi
  • Organizer: Quantum dilogarithm and quantum Teichmuller theory
  • Place of Presentation: Aarhus, Denmark
• [Presentation] Conference : Advanced School and Conference on Homological and Geometrical Methods in Representation theory (2010)
  • Organizer: Periodicities in cluster algebras and cluster categories from quantum groups
  • Place of Presentation: トリエステ
• [Presentation] Dilogarithm identities and cluster algebras (2010)
  • Author(s): T.Nakanishi
  • Organizer: 日本数学会年会
  • Place of Presentation: 慶応大学
• [Presentation] T and Y-systems, dilogarithm identities and cluster algebras : nonsimply laced case (2010)
  • Author(s): R.Inoue, O.lyama, B.Keller, A.Kuniba, T.Nakanishi
  • Organizer: 日本数学会年会
  • Place of Presentation: 慶応大学
• [Presentation] T-systemの周期性について (2009)
  • Author(s): 中西知樹
  • Organizer: 日本数学会年会
  • Place of Presentation: 東京
  • Year and Date: 2009-03-29
• [Presentation] T-systemの周期性について (2009)
  • Author(s): Inoue, Iyama, Kuniba, Suzuki
  • Organizer: 日本数学会年会
  • Place of Presentation: 東大
• [Presentation] Periodicity of T-systems (2009)
  • Organizer: Representation Theories of Algebraic and Quantum Groups
  • Place of Presentation: 和歌山県勝浦市
• [Presentation] T-systems, Y-systems, and cluster algebras (2009)
  • Organizer: Satellite seminar on workshop "Triangulated Categories and Related Topics"
  • Place of Presentation: 名大
• [Presentation] Periodicity of T-systems (2009)
  • Organizer: Infinite Analysis 09-New Trends in Quauntum Integrable Systems
  • Place of Presentation: 京大
• [Presentation] Periodicity of T-systems (2009)
  • Organizer: Integrable systems and their applications
  • Place of Presentation: はこだて未来大
• [Presentation] T-systems, cluster algebras, and cluster categories (2009)
  • Organizer: Topics on q-deformed symmetric polynomilas
  • Place of Presentation: 静岡県裾野市
• [Presentation] T-system の周期性について (2009)
  • Author(s): Inoue, Iyama, Kuniba, Suzuki
  • Organizer: 日本数学会年会
  • Place of Presentation: 東大
• [Presentation] Workshop : Representation Theories of Algebraic and Quantum Groups (2009)
  • Organizer: Periodicity of T-systems
  • Place of Presentation: 和歌山県勝浦市
• [Presentation] Workshop : Satellite seminar on workshop "Triangulated Categories and Related Topics" (2009)
  • Organizer: T-systems, Y-systems, and cluster algebras
  • Place of Presentation: 名大
• [Presentation] Workshop : Infinite Analysis 09-New Trends in Quauntum Integrable Systems (2009)
  • Organizer: Periodicity of T-systems
  • Place of Presentation: 京大
• [Presentation] Workshop : Integrable systems and their applications (2009)
  • Organizer: Periodicity of T-systems
  • Place of Presentation: はこだて未来大
• [Presentation] Workshop : Topics on q-deformed symmetric polynomilas (2009)
  • Organizer: T-systems, cluster algebras, and cluster categories
  • Place of Presentation: 静岡県裾野市
• [Presentation] Q-system, T-system, and Y-system (2008)
  • Author(s): 中西知樹
  • Organizer: Crystals and Tropical Combinatorics
  • Place of Presentation: 京都
  • Year and Date: 2008-08-29
• [Remarks] ホームページ等
• [Remarks]
  • URL: http://www.math.nagoya-u.ac.jp/~nakanisi/index.html