Algebra and geometry of generalised quantum Calogero-Moser problems and applications

广义量子Calogero-Moser问题的代数和几何及其应用

• 批准号: EP/E004008/1
• 负责人: Alexander Veselov
• 金额: $ 28.18万
• 依托单位: Loughborough University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2006
• 资助国家: 英国
• 起止时间: 2006 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FE004008%2F1
• 关键词: Algebra; geometry; generalised; quantum; Calogero

Calogero-Moser systems describe an integrable interaction of particles on the line. They have a natural generalisation related to any Coxeter group due to Olshanetsky and Perelomov. In 1996 Chalykh, Feigin and Veselov discovered that in the quantum case there exist non-symmetric integrable generalisations as well. The nature of these generalisations remained unclear until recent works by Sergeev and Veselov who discovered their relation with the Lie superlagebras and super Jack polynomials.One of the main aims of the project is to develop this relation further to include the theory of symmetric superspaces and super Macdonald polynomials. The proposed investigation of spectral decomposition of the deformed Calogero-Moser operators and their difference analogues is not only of great interest by itself but also will have very important applications in representation theory of Lie superalgebras.Another direction of the proposed research is the investigation and classification of various generalisations of Coxeter root systems appeared in relation with quantum Calogero-Moser problems. This can be applied to other areas of Mathematics and Theoretical Physics including the theory of Huygens' principle, Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation and Frobenius manifolds.

Calogero-Moser系统描述了线上粒子的可积相互作用。他们有一个自然的概括有关的任何Coxeter组由于Olshanetsky和Perelomov。1996年，Chalykh、Feigin和Veselov发现在量子情形下也存在非对称可积推广。这些推广的性质仍然不清楚，直到最近的工作谢尔盖耶夫和Veselov谁发现他们的关系与李超lagebras和超级杰克多项式。该项目的主要目的之一是发展这种关系进一步包括理论的对称超空间和超级麦克唐纳多项式。变形Calogero-Moser算子及其差分类似算子谱分解的研究不仅本身具有重要意义，而且在李超代数表示论中也有重要的应用，研究的另一个方向是对与量子Calogero-Moser问题有关的Coxeter根系的各种推广的研究和分类.这可以应用于数学和理论物理的其他领域，包括惠更斯原理，Witten-Dijkgraaf-Verlinde-Verlinde（WDVV）方程和Frobenius流形。

项目成果

Polynomial Solutions of the Knizhnik-Zamolodchikov Equations and Schur-Weyl Duality

Knizhnik-Zamolodchikov 方程和 Schur-Weyl 对偶的多项式解

• DOI: 10.1093/imrn/rnm046
• 发表时间: 2007
• 期刊: INTERNATIONAL MATHEMATICS RESEARCH NOTICES
• 影响因子: 1
• 作者: Felder Giovanni

Whittaker-Hill equation and semifinite-gap Schroedinger operators

Whittaker-Hill 方程和半有限隙薛定谔算子

• DOI: 10.48550/arxiv.0906.1697
• 发表时间: 2009
• 作者: Hemery A

On the rational monodromy-free potentials with sextic growth

论有性增长的理性无单性势

• DOI: 10.1063/1.3001604
• 发表时间: 2009
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Gibbons J

On Quadrirational Yang-Baxter Maps

关于四边形 Yang-Baxter 映射

• DOI: 10.48550/arxiv.0911.2895
• 发表时间: 2009
• 作者: Papageorgiou V

Whittaker-Hill equation and semifinite-gap Schrödinger operators

Whittaker-Hill 方程和半有限隙薛定谔算子

• DOI: 10.1063/1.3455367
• 发表时间: 2010
• 期刊: Journal of Mathematical Physics
• 影响因子: 1.3
• 作者: Hemery A