2007 Fiscal Year Final Research Report Summary
Integrable Systems and Combinatorial Representation Theory
| Project/Area Number |
18540030
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Osaka University |
|---|
Principal Investigator |
OKADO Masato Osaka University, Graduate School of Engineering Sience, Associate Professor (70221843)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
YAMADA Yasuhiko Kobe Univ, Grad Sch. of Science, Professor (00202383)
KUNIBA Atsuo Univ. of Tokyo, Grad Sch. of Genera Arts, Associate Professor (70211886)
NOBE Atsushi OSAKA UNIVERSITY, Graduate School of Engineering Sience, Assistant Professor (80397728)
|
|---|
| Project Period (FY) |
2006 – 2007
|
| Keywords | Integrable system / Quantum group / Yang-Baxter eouation |
|---|
| Research Abstract |
During the period of research project, we mainly obtained the following results.
1. [Affine geometric crystal]
In collaboration with M. Kashiwara and T. Nakashima, we constructed geometric crystals associated to nonexceptional affine Lie algebras. We confirmed that the ultra-discrete limit of these geometric crystals coincide with the limit of previously known perfect crystals. Moreover, except type C, we obtained explicit formulas for birational maps, called tropical R maps, that satisfy the Yang-Baxter equation.
2. [Existence of crystal bases of the KR modules for nonexceptional types]
There was a conjecture saying that any finite-dimensional representation of a quantum affirm algebra that has an integer multiple of a level 0 fundamental weight as highest weight (KR module) has a crystal base. We solved this conjecture for all affine Lie algebras of nonexceptional types. In collaboration with A. Schilling, we also proved that the crystals of type B^<(1)>_n, D^<(1)>_n, and A^<(2)>_<2n-1> are isomorphic to the combinatorial crystals recently constructed by Schilling.
3. [Construction of the coherent family of perfect crystals for exceptional types]
In collaboration with M. Kashiwara, K.C. Misra and D. Yamada, we revealed the crystal structure of the perfect crystals associated to the exceptional affine lie algebra D^<(3)>_4 at any level.
|
