Algebraic analysis of infinite symmetry

无限对称的代数分析

• 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; algebraic analysis; infinite symmetry; solvable model; quantum group; crystal base; correlation function

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，具有负水平的最高权重模块的不可减至特征是由Kazhdan-Lusztig多项式给出的。在整体级别的情况下已证明了这一猜想。我们成功证明了在非综合案例中的猜想。这使我们证明了卢斯蒂格在雪瓦利组的模块化表示方面的另一个猜想。2。晶体碱基的研究我们成功地构建了从任意有限维表示的量子代数代数的FOCK表示，具有完美的晶体。这是从| q | << 1。此外，我们证明该FOCK表示将量子仿射代数的张量和不可约最高模块分解为量子仿射代数的张量。这些结果最近适用于年轻法语数学的模块化对称性组的研究。可解决模型的研究q-kz方程的解决方案在其参数q满足| q | <1时已知。当| q | = 1时，我们构建了其解决方案并研究了其属性。该解决方案被认为是无差距制度中XXZ模型的相关函数，我们期望它的进一步发展。4。顶点操作员顶点操作员的研究是量子仿射代数有限维代表索引的不可还原最高权重模块的操作员。我们通过晶体碱基分析顶点操作员，并导致量子仿射代数的新表示。