| Project/Area Number |
14540006
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | University of Tsukuba |
|---|
Principal Investigator |
NAITO Satoshi University of Tsukuba, Graduate School of Pure and Applied Sciences, Associate Professor, 大学院・数理物質科学研究科, 助教授 (60252160)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
TAKEUCHI Mitsuhiro University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (00015950)
MORITA Jun University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (20166416)
MIYAMOTO Masahiko University of Tsukuba, Graduate School of Pure and Applied Sciences, Professor, 大学院・数理物質科学研究科, 教授 (30125356)
SAGAKI Daisuke University of Tsukuba, Graduate School of Pure and Applied Sciences, Research Associate, 大学院・数理物質科学研究科, 助手 (40344866)
SAITO Yoshihisa The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (20294522)
|
|---|
| Project Period (FY) |
2002 – 2004
|
| Project Status |
Completed (Fiscal Year 2004)
|
| Budget Amount *help |
¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2004: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2003: ¥1,400,000 (Direct Cost: ¥1,400,000)
Fiscal Year 2002: ¥1,700,000 (Direct Cost: ¥1,700,000)
|
|---|
| Keywords | crystal base / quantum affine algebra / path model / extremal weight module / standard module / Lakshmibai-Seshadri path / level zero / fundamental representation |
|---|
| Research Abstract |
M.Kashiwara introduced the notion of an extremal weight module $V(lambda)$ with an integral weight $lambda$ as an extremal weight over the quantum group $U_q(g)$ associated to a (general) Kac-Moody algebra $g$, by generalizing the notion of an integrable highest weight module. When $g$ is an affine Lie algebra and an integral weight $lambda$ is of level zero, we obtain certain finite-dimensional representations of a quantum affine algebra as quotient modules of the extremal weight module $V(lambda)$. For example, we obtain standard modules $M(lambda)$ introduced by H.Nakajima by an algebro-geometric method, which have turned out to coincide with quantum Weyl modules $M(lambda)$ introduced by V.Chari and A.Pressley. In our series of works from 2002 to 2004, for an integral weight $lambda$ of level zero that is a sum of level-zero fundamental weights for the affine Lie algebra $g$, we studied a certain crystal $B(lambda)_{cl}$, which is (modulo the null root of $g$) the crystal of all Lakshmibai-Seshadri paths of shape $lambda$. As a result, we proved that this crystal $B(lambda)_{cl}$ is isomorphic as a crystal to a tensor product of the path models for level-zero fundamental representations of the quantum affine algebra. From this result, we see that the crystal $B(lambda)_{cl}$ can be thought of as a path model for the standard module $M(lambda)$. Moreover, using our results above, we can describe the branching rule of the standard module $M(lambda)$ over the quantum affine algebra with respect to the restriction to the quantum group $U_q(g_0)$ associated to a (canonical) finite-dimensional reductive Lie subalgebra $g_0$ of $g$ in a combinatorial way, by using elements of the Weyl group $W$ of $g$ and the Bruhat order on $W$.
|
