Calogero-Moser systems, Cherednik algebras and Frobenius structures

Calogero-Moser 系统、Cherednik 代数和 Frobenius 结构

• 批准号: EP/F032889/1
• 负责人: Misha Feigin
• 金额: $ 36.42万
• 依托单位: University of Glasgow
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2008
• 资助国家: 英国
• 起止时间: 2008 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FF032889%2F1
• 关键词: Calogero; Moser; systems; Cherednik; algebras

The proposed project lies in the areas of integrable systems, and more specifically Calogero-Moser systems, Cherednik algebras and the theory of Frobenius manifolds. The Calogero-Moser system is one of the most important integrable systems because of its far reaching and deep connections with algebra, geometry, representation theory and other branches of mathematics and mathematical physics. The Cherednik algebra is a remarkable algebra which has been extensively studied in the last decade. It appeared as a powerful tool used to solve problems in combinatorics but it also has deep connections with geometry. The representation theory of Cherednik algebras is a very active, rapidly developing area. Certain beautiful parts of the theory have already been constructed. Cherednik algebras are connected with Calogero-Moser operators through the faithful representation of Cherednik algebras by Dunkl operators.Frobenius manifolds were introduced by Dubrovin in the 90's. He formalized in a geometrical way the associativity conditions from topological field theories also known as Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. The theory of quantum Calogero-Moser problems, Seiberg-Witten theory and Frobenius manifolds coming from singularity theory give rise to remarkably simple solutions of the WDVV equations.The aim of the project is to extend considerably and deepen the above mentioned connections between the areas as well as to develop new perspectives. The generalised quantum integrable Calogero-Moser problems will be obtained from special modules of Cherednik algebras. In this way we expect to recover the integrability of some of the known systems as well as to discover new integrable systems. This will also give a unified approach to the integrability of generalised Calogero-Moser systems. We are going to study special solutions of the WDVV equations connected to trigonometric Calogero-Moser problems and the Frobenius manifolds on the orbit spaces of extended affine Weyl groups. The project will develop the differential geometry on the discriminant submanifolds in the Frobenius manifolds coming from Coxeter groups. Finally, we intend to introduce and develop new connections between Frobenius structures on these orbit spaces and special modules for the Cherednik algebras. We also plan to explore this new connection and develop its consequences for both Frobenius manifolds and Cherednik algebras.

拟议的项目属于可积系统领域，更具体地说是Calogero-Moser系统、Cherednik代数和Frobenius流形理论。 Calogero-Moser 系统是最重要的可积系统之一，因为它与代数、几何、表示论以及数学和数学物理的其他分支有着深远而深刻的联系。切雷德尼克代数是一个非凡的代数，在过去十年中得到了广泛的研究。它似乎是解决组合数学问题的强大工具，但它也与几何学有着深厚的联系。 Cherednik 代数的表示论是一个非常活跃、快速发展的领域。该理论的某些优美部分已经被构建出来。 Cherednik 代数通过 Dunkl 算子忠实地表示 Cherednik 代数与 Calogero-Moser 算子联系起来。Frobenius 流形由 Dubrovin 在 90 年代引入。他以几何方式形式化了拓扑场论中的结合性条件，也称为 Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) 方程。量子 Calogero-Moser 问题的理论、Seiberg-Witten 理论和来自奇点理论的 Frobenius 流形产生了 WDVV 方程的非常简单的解。该项目的目的是大大扩展和加深上述领域之间的联系：以及开拓新的视角。广义量子可积Calogero-Moser 问题将从Cherednik 代数的特殊模块中获得。通过这种方式，我们期望恢复一些已知系统的可积性，并发现新的可积系统。这也将为通用 Calogero-Moser 系统的可集成性提供统一的方法。我们将研究与三角 Calogero-Moser 问题相关的 WDVV 方程的特殊解以及扩展仿射 Weyl 群轨道空间上的 Frobenius 流形。该项目将开发来自 Coxeter 群的 Frobenius 流形中判别子流形的微分几何。最后，我们打算引入和开发这些轨道空间上的 Frobenius 结构与 Cherednik 代数的特殊模块之间的新联系。我们还计划探索这种新的联系，并研究其对 Frobenius 流形和 Cherednik 代数的影响。

项目成果

A class of Baker-Akhiezer arrangements

一类 Baker-Akhiezer 安排

• DOI: 10.48550/arxiv.1212.3597
• 发表时间: 2012
• 作者: Feigin M

Singular polynomials from orbit spaces

轨道空间的奇异多项式

• DOI: 10.48550/arxiv.1110.1946
• 发表时间: 2011
• 作者: Feigin M

Generalized Calogero-Moser systems from rational Cherednik algebras

来自有理 Cherednik 代数的广义 Calogero-Moser 系统

• DOI: 10.1007/s00029-011-0074-y
• 发表时间: 2011
• 期刊: Selecta Mathematica
• 作者: Feigin M

Generalized Macdonald-Ruijsenaars systems

广义麦克唐纳-瑞森纳尔系统

• DOI: 10.48550/arxiv.1102.3903
• 发表时间: 2011
• 作者: Feigin M

Trigonometric Solutions of WDVV Equations and Generalized Calogero-Moser-Sutherland Systems

WDVV 方程和广义 Calogero-Moser-Sutherland 系统的三角解

• DOI: 10.3842/sigma.2009.088
• 发表时间: 2009
• 期刊: Methods and Applications
• 作者: Feigin M