2004 Fiscal Year Final Research Report Summary
Combinatorial Study of Crystal Bases and its Application to Discrete Integrable Systems
| Project/Area Number |
14540026
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | OSAKA UNIVERSITY |
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Principal Investigator |
OKADO Masato Osaka University, Graduate School of Engineering Science, Associate Professor, 大学院・基礎工学研究科, 助教授 (70221843)
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| Co-Investigator(Kenkyū-buntansha) |
OGAWA Toshiyuki Osaka University, Graduate School of Engineering Science, Associate Professor, 大学院・基礎工学研究科, 助教授 (80211811)
NOBE Atsushi Osaka University, Graduate School of Engineering Science, Research Assistant, 大学院・基礎工学研究科, 助手 (80397728)
KUNIBA Atsuo Univ of Tokyo, Grad Sch of General Arts, Associate Professor, 大学院・総合文化研究科, 助教授 (70211886)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Integrable system / Quantum group / Yang-Baxter equation / Cell automaton |
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| Research Abstract |
(i) Geometric crystal
Geometric crystal, indtoduced axiomatically by Berenstein and Kazhdan, was consttructed explicitly for type D^(1)_n of Kac-Moody Lie algebras. By using a matrix realization of this geometric crystal, we constructed a birational mapping on the product (tropical R) commuting with the action of the geometric crystal and also,showed that it satisfies the Yang-Baxter equation. It is believed that a geometric crystal exists arrogated to each vertex of the Dynkin diagram corresponding to the Lie algebra. For type D, there are n vertices, so it is expected that there are n distinct geometric crystals. During the last yeah we calculated the geometric crystal for k=2 by Mathematics in collaboration with Masaki Kashiwara at RIMS, Kyoto University. Data is huge and no way to print it out. However to write it down in a meaningful manner is, besides with the extension to the case when k is greater than 2, becomes a future problem.
(ii)Crystal and soliton cellular automaton associated to an exceptional acne Lie algebra
Coordinate representation of a series of finite crystals for an exceptional affine Lie algebra D_4^(3) is given and the zero action is explicitly obtained. Moreover we constructed a cell automaton corresponding to this series of crystals and determined the internal degree of freedom of the solitons appearing in the system and the scattering rule of two solitons.
(iii)Box-ball system with reflecting end
We have extended the box-ball system, an important example of ultra discrete integrable systems, to the case with one reflecting end. Similar to the usual box-ball system, it has an infinite family of commuting time evolutions and conserved quantities associated to each time evolution. We also defined soliton states and described the reflection rule of one soliton and the scattering rule of two solitons in terms of combinatorics of crystals.
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