Method of Algebraic Analysis in Mathematical Physics (with the emphasis on Representation Theory, Combinatorics and Complex Analysis)

• Project/Area Number: 17340038
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyoto University
• Principal Investigator: MIWA Tetsuji Kyoto University, 大学院・理学研究科, 教授 (10027386)
• Co-Investigator(Kenkyū-buntansha): JIMBO Michio 立教大学, 理学部, 教授 (80109082) ; OKADO Masato 大阪大学, 基礎工学研究科, 准教授 (70221843) ; NAKAYASHIKI Atsushi 九州大学, 数理学研究院, 准教授 (10237456) ; TAKEYAMA Yoshihiro 筑波大学, 数理物質科学研究科, 講師 (60375392) ; IOHARA Kenji リョン大学(フランス), 数学教室, 准教授 (00322199) ; SUZUKI Takeshi 岡山大学, 自然科学研究科, 准教授 (30335294) ; TAKEMURA Kouichi 横浜市立大学, 国際総合科学研究科, 准教授 (10326069)
• Project Period (FY): 2005 – 2008
• Project Status: Completed (Fiscal Year 2008)
• Budget Amount: ¥9,240,000 (Direct Cost: ¥8,100,000、Indirect Cost: ¥1,140,000); Fiscal Year 2008: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2007: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2006: ¥1,900,000 (Direct Cost: ¥1,900,000); Fiscal Year 2005: ¥2,400,000 (Direct Cost: ¥2,400,000)
• Keywords: 表現論 / 組み合わせ論 / 可解模型 / XXZ模型 / 真空期待値 / 有限温度 / フェルミオン公式 / Whittackerベクトル / 量子戸田方程式 / 量子戸田格子 / Wittackerベクトル / 自由fermion / qKZ方程式 / 自由場表示 / Kirillov-Reshetikhin加群 / 結晶基底 / 量子スピン系 / 相関関数 / 密度行列 / XXX模型 / Virasoro代数 / 極小表現 / 単項基底 / 融合積

Research Abstract

数学を形作る第一の要素は数であり、その次ぎにくるのは函数である。函数は数に数を対応させるものである。もう一段上になって、函数に函数を対応させるものを作用素という。作用素は2つのものから第3のものを、順に続けることによって作り出すことができるが、順番を変えると結果が異なる。これを作用素の非可換性という。非可換な作用素がどのような等式によって統制されるかを研究するのが代数解析である。本研究では、磁石のような物理系を数学の言葉で作用素を用いてとらえ、その非可換性を研究することによって温度変化する磁化の強さのような函数を決定し、物理系を特徴づける数にまで迫ろうというものである。

Research Products

• [Journal Article] Hidden Grassmann structure in the XXZ model. II. Creation operators (2009)
  • Author(s): Boos, H.; Jimbo, M.; Miwa, T.; Smirnov, F.; Takeyama, Y
  • Journal Title: Comm. Math. Phys 286, no.3 Pages: 875-932
  • Peer Reviewed
• [Journal Article] Hidden Grassmann Structure in the XXZ Model II : Creation Operators (2009)
  • Author(s): H. Boos, M. Jimbo, T. Miwa, E. Smirnov, Y. Takeyama
  • Journal Title: Comm. Math. hys. 286 Pages: 875-932
  • Peer Reviewed
• [Journal Article] Principal $\widehat{\mathfrak{sl}_3$ subspaces and quantum Toda Hamiltonian (2009)
  • Author(s): B. Feigin, E. Feigin, M. jimbo, T. Miwa, E. Mukhin
  • Journal Title: Advanced Studies in Pure Mathematics 54 Pages: 109-166
  • Peer Reviewed
• [Journal Article] Affine geometric crystals and limit of perfect crystals (2008)
  • Author(s): Kashiwara, Masaki, Nakashima, Toshiki, Okado, Masato
  • Journal Title: Trans. Amer. Math. Soc 360, no.7 Pages: 3645-3686
  • Peer Reviewed
• [Journal Article] A $\phi\sb {1,3}$-filtration of the Virasoro minimal series M(p,p') with 1<p'/p<2" (2008)
  • Author(s): Feigin, B., Feigin, E., Jimbo, M., Miwa, T., Takeyama, Y
  • Journal Title: Publ. Res. Inst. Math. Sci 44, no.2 Pages: 213-257
  • Peer Reviewed
• [Journal Article] Schilling, Anne Existence of Kirillov-Reshetikhin crystals for nonexceptional types (2008)
  • Author(s): Okado, Masato
  • Journal Title: Represent 12 Pages: 186-207
  • Peer Reviewed
• [Journal Article] Modules de plus haut poids unitarisables sur la super-algebre de Virasoro $N=2$ tordue. (French)[Unitarizable highest-weight modules on the twisted $N=2$ super Virasoro algebra] (2008)
  • Author(s): Iohara, Kenji
  • Journal Title: Ann. Inst 58, no.3 Pages: 733-754
  • Peer Reviewed
• [Journal Article] Differential structure of abelian functions (2008)
  • Author(s): Cho, Koji, Nakayashiki, Atsushi
  • Journal Title: Internat. J. Math 19, no.2 Pages: 145-171
  • Peer Reviewed
• [Journal Article] Sets of rigged paths with Virasoro characters (2008)
  • Author(s): Feigin, B., Jimbo, M., Miwa, T., Mukhin, E., Takeyama, Y
  • Journal Title: Ramanujan J 15, no.1 Pages: 123-145
  • Peer Reviewed
• [Journal Article] The best constant of Sobolev inequality on a bounded interval (2008)
  • Author(s): Watanabe, K., Kametaka, Y., Nagai, A., Takemura, K., Yamagishi, H
  • Journal Title: J. Math. Anal. Appl 340, no.1 Pages: 699-706
  • Peer Reviewed
• [Journal Article] Free field approach to solutions of the quantum Knizhnik-Zamolodchikov equations. SIGMA Symmetry Integrability (2008)
  • Author(s): Kuroki, Kazunori, Nakayashiki, Atsushi
  • Journal Title: Geom Methods Appl 4(049) Pages: 13-13
  • Peer Reviewed
• [Journal Article] Remarks on the confluent KZ equation for $\germ{sl}\sb 2$ and quantum Painleve equations (2007)
  • Author(s): Jimbo, M, Nagoya, H., Sun, J
  • Journal Title: J. Phys A41, no.17
  • Peer Reviewed
• [Journal Article] Existence of crystal bases for Kirillov-Reshetikhin modules of type $D$ (2007)
  • Author(s): Okado, Masato
  • Journal Title: Publ. Res. Inst. Math. Sci 43, no.4 Pages: 977-1004
  • Peer Reviewed
• [Journal Article] Hidden Grassmann structure in the XXZ model (2007)
  • Author(s): Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y
  • Journal Title: Comm Math. Phys 272, no.1 Pages: 263-281
  • Peer Reviewed
• [Journal Article] Perfect crystals for $U\sb q(D\sp {(3)}\sb 4)$ (2007)
  • Author(s): Kashiwara, M., Misra, K. C., Okado, M., Yamada, D
  • Journal Title: J. Algebra 317, no.1 Pages: 392-423
  • Peer Reviewed
• [Journal Article] Hidden Grassmann structure in the XXZ model (2007)
  • Author(s): T. Miwa, M. Jimbo, Y. Takeyama, 他2名
  • Journal Title: Comm. Math. Phys. 272 Pages: 263-281
  • Peer Reviewed
• [Journal Article] Existence of crystal bases for Kirillov-Reshetikhin modules of type D (2007)
  • Author(s): M. Okado
  • Journal Title: Publ. RIMS 43 Pages: 977-1004
  • Peer Reviewed
• [Journal Article] The quantum Knizhnik-Zamolodchikov equation and non-symmetric Macdonald polynomials (2007)
  • Author(s): M. Kasatani and Y. Takeyama
  • Journal Title: Funkcialaj Ekvacioj 50-3 Pages: 491-509
  • NAID: 130000140696
  • Peer Reviewed
• [Journal Article] Coincident root loci and Jack and Macdonald polynomials for special values of the parameters (2006)
  • Author(s): Kasatani, M., Miwa, T., Sergeev, A. N., Veselov, A. P
  • Journal Title: Contemp. Math 417 Pages: 207-225
  • Peer Reviewed
• [Journal Article] Algebraic representation of correlation functions in integrable spin chains (2006)
  • Author(s): Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y
  • Journal Title: Ann. Henri Poincare 7, no.7-8 Pages: 1395-1428
  • Peer Reviewed
• [Journal Article] Double affine Hecke algebras, conformal coinvariants and Kostka polynomials (2006)
  • Author(s): Suzuki, Takeshi
  • Journal Title: C. R. Math. Acad. Sci 343, no.6 Pages: 383-386
  • Peer Reviewed
• [Journal Article] Density matrix of a finite sub-chain of the Heisenberg anti-ferromagnet (2006)
  • Author(s): Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y
  • Journal Title: Lett Math. Phys 75, no.3 Pages: 201-208
  • Peer Reviewed
• [Journal Article] Reduced qKZ equation and correlation functions of the XXZ model (2006)
  • Author(s): Boos, H. ; Jimbo, M. ; Miwa, T. ; Smirnov, F. ; Takeyama, Y
  • Journal Title: CommMath. Phys 261, no.1 Pages: 245-276
  • Peer Reviewed
• [Journal Article] Algebraic representation of correlation functions in integrable spin chains (2006)
  • Author(s): Boos, H, Jimbo, M, Miwa, T. et al.
  • Journal Title: Ann. Henri Poincare 7・7-8 Pages: 1395-1428
• [Journal Article] Density matrix of a finite sub-chain of the Heisenberg anti-ferromagnet (2006)
  • Author(s): Boos, H, Jimbo, M, Miwa, T. et al.
  • Journal Title: Lett. Math. Phys. 75・3 Pages: 201-208
• [Journal Article] Reduced qKZ equation and correlation functions of the $XXZ$ model (2006)
  • Author(s): Boos, H, Jimbo, M, Miwa, T. et al.
  • Journal Title: Comm. Math. Phys. 261・1 Pages: 245-276
• [Journal Article] Tilting equivalence for superconformal algebras (2006)
  • Author(s): Iohara, Kenji, Koga, Y.
  • Journal Title: Math. Scand. 99・1 Pages: 17-52
• [Journal Article] Crystal interpretation of Kerov-Kirillov-Reshetikhin bijection (2006)
  • Author(s): Kuniba, Atsuo, Okado, Masato, et al.
  • Journal Title: Nuclear Phys. B 740・3 Pages: 299-327
• [Journal Article] Double affine Hecke algebras, conformal coinvariants and Kostka polynomials (2006)
  • Author(s): Suzuki, Takeshi
  • Journal Title: C. R. Math. Acad. Sci. Paris 343・6 Pages: 383-386
• [Journal Article] Reduced qKZ equation and correlation functions of the $XXZ$ model (2006)
  • Author(s): M.Jimbo, T.Miwa et al.
  • Journal Title: Comm.Math.Phys. 261-1
• [Journal Article] Traces on the Sklyanin algebra and correlation functions of the eight-vertex model (2005)
  • Author(s): Boos, H., Jimbo, M., Miwa, T., Smirnov, F., Takeyama, Y
  • Journal Title: J. Phys A 38, no.35 Pages: 7629-7659
  • Peer Reviewed
• [Journal Article] A monomial basis for the Virasoro minimal series M(p,p') : the case 1<p'/p<2 (2005)
  • Author(s): Feigin, B., Jimbo, M., Miwa, T., Mukhin, E., Takeyama, Y
  • Journal Title: Comm Math. Phys 257, no.2 Pages: 395-423
  • Peer Reviewed
• [Journal Article] Rational and trigonometric degeneration of the double affine Hecke algebra of type $\scr A$. (2005)
  • Author(s): Suzuki, Takeshi
  • Journal Title: Int Math. Res. Not no. 37 Pages: 2249-2262
  • Peer Reviewed
• [Journal Article] A monomial basis for the Virasoro minimal series $M(p,p')$ : the case $1<p'/p<2$ (2005)
  • Author(s): Jimbo, M., Miwa, T.et al.
  • Journal Title: Comm.Math.Phys. 257・2 Pages: 395-423
• [Journal Article] Traces on the Sklyanin algebra and correlation functions of the eight-vertex model (2005)
  • Author(s): Jimbo, M., Miwa, T.et al.
  • Journal Title: J.Phys.A 38-35 Pages: 7629-7659
• [Journal Article] Density matrix of a finite sub-chain of the Heisenberg anti-ferromagnet (2005)
  • Author(s): Boos, H, M.Jimbo et al.
  • Journal Title: Lett.Math.Phys. 75 Pages: 201-208
• [Journal Article] A recursion formula for the correlation functions of an inhomogeneous $XXX$ model (2005)
  • Author(s): Feigin, B., Jimbo, M.et al.
  • Journal Title: Algebra i Analiz 17-1 Pages: 115-159
• [Presentation] Correlation functions in the XXZ model (2009)
  • Author(s): Tetsuji Miwa
  • Organizer: Mathematical Physics, from XX to XXI centuries
  • Place of Presentation: Univ. of Lausanne, Switzerland
  • Year and Date: 2009-03-16
• [Presentation] Mathematical Physics : from 20 to 21 centuries In Honor of the 75^<th> Birthday of Ludwig Faddeev (2009)
  • Organizer: Correlation functions of the XXZ model
• [Presentation] XXZ模型のフェルミ粒子描像 (2008)
  • Author(s): 三輪哲二
  • Organizer: 日本数学会・日本物理学会合同講演会
  • Place of Presentation: 近畿大学
  • Year and Date: 2008-03-22
• [Presentation] Galileo Galilei Institute Workshop Low-dimensional Quantum Field Theories and Application (2008)
  • Organizer: Fermion Structure in XXZ model
  • Place of Presentation: Fermion Structure in XXZ model