Study of solutions to the reflection equation associated with dynamical Yang-Baxter maps

• Project/Area Number: 26400031
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Hokkaido University
• Principal Investigator: SHIBUKAWA Youichi 北海道大学, 理学研究院, 准教授 (90241299)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: ダイナミカル・ヤン・バクスター写像 / 反射方程式 / カンドル / 可換作用素

Outline of Final Research Achievements

The main results of this research are: (1) to clarify what the reflection equation associated with dynamical Yang-Baxter maps is; and (2) to construct solutions to the reflection equation associated with the dynamical Yang-Baxter maps from dynamical braces. In addition, we produced dynamical Yang-Baxter maps by means of quandles, which play an important role in the study of links. We showed that these dynamical Yang-Baxter maps give birth to weak Hopf algebras.

Research Products

• [Presentation] ダイナミカル・ヤン・バクスター写像とカンドル (2017)
  • Author(s): 澁川陽一
  • Organizer: Meeting for Study of Number theory, Hopf algebras and related topics
  • Place of Presentation: 富山大学理学部（富山県・富山市）
• [Presentation] Construction of Hopf algebroids by means of parallelepipeds (2016)
  • Author(s): Youichi Shibukawa
  • Organizer: Hopf-Algebra Conference in Tsukuba
  • Place of Presentation: つくば国際会議場（茨城県・つくば市）
  • Int'l Joint Research / Invited
• [Presentation] Hopf algebroids and dynamical Yang-Baxter maps (2015)
  • Author(s): Youichi Shibukawa
  • Organizer: Tsukuba Mini-conference on Hopf algebras and differential Galois theory
  • Place of Presentation: 筑波大学（茨城県・つくば市）
  • Year and Date: 2015-09-14
  • Invited
• [Presentation] ダイナミカル・ヤン・バクスター写像を用いたホップ亜代数の構成 (2015)
  • Author(s): 澁川陽一
  • Organizer: 東京無限可積分系セミナー
  • Place of Presentation: 東京大学（東京都目黒区）
  • Year and Date: 2015-01-22
• [Presentation] Hopf algebroids associated with dynamical Yang-Baxter maps (2014)
  • Author(s): 澁川陽一
  • Organizer: 日本数学会2014年度秋季総合分科会
  • Place of Presentation: 広島大学（広島県東広島市）
  • Year and Date: 2014-09-25
• [Presentation] Rigid tensor categories associated with dynamical Yang-Baxter maps (2014)
  • Author(s): 澁川陽一
  • Organizer: 日本数学会2014年度秋季総合分科会
  • Place of Presentation: 広島大学（広島県東広島市）
  • Year and Date: 2014-09-25