Analysis of the solution space for the quantum KZ equation and its applications to integrable systems

• Project/Area Number: 23740119
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Global analysis
• Research Institution: University of Tsukuba
• Principal Investigator: TAKEYAMA Yoshihiro 筑波大学, 数理物質系, 准教授 (60375392)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
• Keywords: 可積分系 / 多重ゼータ値 / 差分方程式 / 量子KZ方程式 / 二重アフィンヘッケ代数

Research Abstract

We get two results about related problems to the quantum Knizhnik-Zamolodchikov equation. First, we constructed an algebra which describes the multiplication structure of a family of q-series containing a q-analogue of multiple zeta values. A family of linear relations called the double shuffle relations is formulated and proved in our framework. Second, we defined a discrete analogue of the Hamiltonian of the non-ideal Bose gas with delta-potentials, and constructed eigenfunctions by means of the Bethe ansatz method making use of a representation of the affine Hecke algebra.

Research Products

• [Journal Article] A discrete analogue of periodic delta Bose gas and affine Hecke algebra (2014)
  • Author(s): Yoshihiro Takeyama
  • Journal Title: FUNKCIALAJ EKVACIOJ Volume: 57 Pages: 107-118
  • NAID: 130003391676
  • URL: http://www.math.sci.kobe-u.ac.jp/~fe/xml/fe57-1-5.xml
  • Peer Reviewed
• [Journal Article] The Algebra of a q-Analogue of Multiple Harmonic Series (2013)
  • Author(s): Yoshihiro Takeyama
  • Journal Title: Symmetry, Integrability and Geometry : Methods and Applications (SIGMA) Volume: 9 Pages: 15-15
  • DOI: 10.3842/sigma.2013.061
  • NAID: 120007136785
  • Peer Reviewed
• [Journal Article] A restricted sum formula for a q-analogue of multiple zeta values (2013)
  • Author(s): Yoshihiro Takeyama
  • Journal Title: Symmetries, Integrable Systems and Representations (Springer Proceedings in Mathematics & Statistics) Volume: 40 Pages: 561-573
  • DOI: 10.1007/978-1-4471-4863-0_24
  • ISBN: 9781447148623, 9781447148630
  • Peer Reviewed
• [Journal Article] Quadratic relations for a q-analogue of multiple zeta values (2011)
  • Author(s): Yoshihiro Takeyama
  • Journal Title: Ramanujan J. Volume: 27(1) Issue: 1 Pages: 15-18
  • DOI: 10.1007/s11139-011-9328-8
  • Peer Reviewed
• [Presentation] A discrete analogue of periodic delta Bose gas and affine Hecke algebra (2014)
  • Author(s): 竹山美宏
  • Organizer: 日本数学会年会
  • Place of Presentation: 学習院大学
  • Year and Date: 2014-03-17
• [Presentation] JSPS-NWO Seminar : Analysis (2013)
  • Author(s): 竹山美宏
  • Organizer: Geometry and Group Representations for Homogeneous Spaces
  • Place of Presentation: 名古屋大学
  • Year and Date: 2013-08-27
• [Presentation] A generalization of duality for finite multiple harmonic sum (2013)
  • Author(s): 竹山美宏
  • Organizer: Infinite Analysis : Past, Present and Future
  • Place of Presentation: 京都大学
  • Year and Date: 2013-03-06
• [Presentation] On relations for a q-analogue of multiple zeta values (2012)
  • Author(s): Yoshihiro Takeyama
  • Organizer: Infinite Analysis 11 - Frontier of Integrability -(招待講演)
  • Place of Presentation: 東京大学
• [Presentation] On relations for a q-analogue of multiple zeta values (2011)
  • Author(s): 竹山美宏
  • Organizer: Infinite Analysis 11 - Frontier of Integrability -
  • Place of Presentation: 東京大学
  • Year and Date: 2011-07-28
• [Presentation] A discrete analogue of periodic delta Bose gas and affine Hecke algebra
  • Author(s): Yoshihiro Takeyama
  • Organizer: JSPS-NWO Seminar: Analysis, Geometry and Group Representations for Homogeneous Spaces
  • Place of Presentation: 名古屋大学 (愛知県名古屋市千種区)
  • Invited
• [Presentation] A discrete analogue of periodic delta Bose gas and affine Hecke algebra
  • Author(s): 竹山美宏
  • Organizer: 日本数学会年会
  • Place of Presentation: 学習院大学 (東京都豊島区)