2007 Fiscal Year Final Research Report Summary
A quasi-triangular structure in Kashiwara-Miwa model
| Project/Area Number |
17540204
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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 |
Global analysis
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| Research Institution | Rikkyo University |
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Principal Investigator |
YAMADA Yuji Rikkyo University, Dep. of math., lecturer (40287917)
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| Co-Investigator(Kenkyū-buntansha) |
SHIRAISHI Junichi Univ. of Tokyo, Graduate school of math., Sciences associate professor (20272536)
KAKEI Saboro Rikkyo Univ., Dep. of math., associate professor (60318798)
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| Project Period (FY) |
2005 – 2007
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| Keywords | Yang-Baxter equation / reflection equation / quasi-triangular Hopf algebra |
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| Research Abstract |
We study the classification of the solutions to the reflection equation for Cremmer-Gervais R-matrix in N=3 case. The R-matrix of Cremmer-Gervais type is obtained from Uq(sl_N) through the theory of quasi-triangular Hopf algebras of Drinfeld as solutions to the Yang-Baxter equation. There are only two types of the R-matrices which are obatained through the theory of quasi-triangular Hopf algebras. One is the series of Belavin' s R-matrices, and the other is the series of Cremmer-Gervais type. The algebraic structures of the solutions to the reflection equations are not well understood. We have only two cases in which all solutions to the reflection equation are known, (1) N=2 Belavin' s elliptic R-matrix (eight-vertex case), and (2) N=3 trigonomerically degenerated Belavin' s R-matix case. In order to understand the algebraic structure in solutions to the reflection equation, we study the case of N=3 Cremmer-Gervais R-matix case (with Kohei MOTEGI). The N=2 Cremmer-Gervais R-matrix is only the degenerated N=2 Belavin' s elliptic R-matrix.
The solution space of the reflection equation to the N=3 Cremmer-Gervais R-matrix obtained through this study is described by the rational surface in projective spaces. In the case of the N=3 trigonometric R-matrix, the parameter space of the solution of the reflection equation is the Segre three-fols in $P^5(C)$. In our case there appeared two parameters spaces. One is the same Segre three-fold. But the other new parameter space is embedded in the projective space$P^10$.
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