Algebraic analysis of infinite symmetry

无限对称的代数分析

基本信息

批准号：
08454006
负责人：
KASHIWARA Masaki
金额：
$ 3.46万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至无数据
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08454006/
关键词：
algebraic analysis infinite symmetry solvable model quantum group crystal base correlation function

项目摘要

1. Kazhdan-Lusztig Conjecture Lusztig conjectured that the irreducible character of highest weight modules with negative level is given by the Kazhdan-Lusztig polynomials. This conjecture had been proved in the integral level case. We succeeded to prove this conjecture in the non-integral case. This leads us to prove another conjecture of Lusztig on the modular representation of Chevalley groups.2. Study of Crystal bases We succeeded to construct the Fock representations of the quantum affine algebras starting from an arbitrary finite-dimensional representations with perfect crystal. This is obtained as an analytic continuation from |q|<< 1. Furthermore we showed that this Fock representation decomposes to the tensor product of the Boson Fock space and the irreducible highest modules of the quantum affine algebras.These results are recently applied to the study of modular representations of spin symmetric groups by the young French mathematicians Leclerc-Thibon.3. Study of solvable models The solution to the q-KZ equation has been known when its parameter q satisfies |q|< 1. We constructed its solution when |q|=1 and studied its properties. This solution is supposed to be the correlation function of XXZ-model in the gap-less regime and we expect its further development.4. Study of vertex operators Vertex operator is an operator on the irreducible highest weight modules indexed by finite-dimensional representation of quantum affine algebra. We analyze the vertex operator via crystal bases and leads to a new class of representations of quantum affine algebras.

1. Kazhdan-Lusztig 猜想 Lusztig 猜想负水平的最高权重模块的不可约特征由 Kazhdan-Lusztig 多项式给出。这个猜想在积分级情况下得到了证明。我们在非整数情况下成功证明了这个猜想。这使我们证明了Lusztig关于Chevalley群模表示的另一个猜想。 2．晶体基的研究我们成功地从完美晶体的任意有限维表示开始构造了量子仿射代数的福克表示。这是从 |q|<< 1 的解析延拓得到的。此外，我们证明了这种 Fock 表示分解为玻色子 Fock 空间和量子仿射代数的不可约最高模的张量积。这些结果最近被应用于法国青年数学家Leclerc-Thibon对自旋对称群模表示的研究。3．可解模型的研究当q-KZ方程的参数q满足|q|< 1时，q-KZ方程的解是已知的。我们构造了|q|=1时的解并研究了其性质。该解应该是无间隙状态下XXZ模型的相关函数，我们期待它的进一步发展。 4．顶点算子的研究顶点算子是由量子仿射代数的有限维表示索引的不可约最高权模上的算子。我们通过晶体基分析顶点算子，并得出一类新的量子仿射代数表示。