Angular Cherednik Algebras and Integrability

Angular Cherednik 代数和可积性

• 批准号: EP/W013053/1
• 负责人: Misha Feigin
• 金额: $ 52.52万
• 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: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FW013053%2F1
• 关键词: Angular; Cherednik; Algebras; Integrability

Integrable systems describe particle interactions where a great deal of precise information on particles behaviour can be obtained. These relate both to classical mechanical systems, where one is interested in trajectories and conserved quantities, as well as to quantum systems where conserved quantities ultimately help to determine the spectrum. Such situations are rare and they tend to point to important mathematical structures. These structures and concepts effectively ensure that additional properties of particle behaviour can be determined. Thus integrable systems can often have deep relations with algebra and geometry and these links have already been very fruitfully explored in the past. For instance, the celebrated Calogero-Moser system describes pairwise interacting particles on the line with potential inversely proportional to the squared distance between the particles; it is deeply related with geometry of symmetric spaces, algebraic geometry, and with Cherednik algebras, which have flourished in the last two decades.The goal of this intradisciplinary project is to bring together expertise in integrable systems with that in geometric representation theory in order to uncover new integrable systems and related algebraic structures, with further intriguing connections with geometry of singularities and Lie theory. A key object of the project is an angular version of the Calogero-Moser system, which corresponds to motion on a higher-dimensional sphere. The non-commutative algebras appearing in this situation are very poorly understood, and much less studied due to their novelty and greater complexity. We will develop the representation theory of these algebras. Geometrically, these algebras quantize a new class of symplectic singularities. We will study these singular spaces using both geometric and representation theoretic techniques. We expect that this will lead to a beautiful class of examples, uniting symplectic quotient singularities and nilpotent orbit closures, which is a remarkable new illustration of the interplay between geometry and algebra.We will also find, and study, angular versions of the relativistic extensions of Calogero-Moser systems, which are expected to be new integrable systems. The corresponding algebraic structures will be uncovered: they are expected to involve a novel blending of Cherednik algebras and quantum groups which are ubiquitos in many areas of mathematics.

可积系统描述了粒子的相互作用，可以获得大量关于粒子行为的精确信息。这些都涉及到经典力学系统，其中一个感兴趣的是轨迹和守恒量，以及量子系统，其中守恒量最终有助于确定光谱。这种情况很少见，它们往往指向重要的数学结构。这些结构和概念有效地确保了可以确定颗粒行为的附加性质。因此，可积系统往往可以有深刻的关系，代数和几何和这些联系已经非常富有成效的探索在过去。例如，著名的Calogero-Moser系统描述了直线上两两相互作用的粒子，其势与粒子间距离的平方成反比;它与对称空间几何、代数几何和切雷德尼克代数有着密切的联系，这个跨学科项目的目标是将可积系统的专业知识与几何学的专业知识结合起来，表示理论，以揭示新的可积系统和相关的代数结构，与奇点几何和李理论进一步有趣的连接。该项目的一个关键目标是Calogero-Moser系统的角度版本，它对应于高维球体上的运动。在这种情况下出现的非交换代数是非常少的理解，更少的研究，由于其新奇和更大的复杂性。我们将发展这些代数的表示论。在几何上，这些代数是一类新的辛奇点。我们将使用几何和表示论的技巧来研究这些奇异空间。我们希望这将导致一个美丽的一类的例子，统一辛商奇点和幂零轨道闭包，这是一个显着的新的说明几何和代数之间的相互作用。我们还将发现，并研究，角版本的相对论性扩展的Calogero-Moser系统，这是预期的新的可积系统。相应的代数结构将被揭示：它们预计将涉及一种新的混合的切雷德尼克代数和量子群，这是无处不在的许多领域的数学。

项目成果

Coulomb branches have symplectic singularities

库仑支具有辛奇点

• DOI: 10.1007/s11005-023-01724-5
• 发表时间: 2023
• 期刊: Letters in Mathematical Physics
• 影响因子: 1.2
• 作者: Bellamy G

Integral expressions for derivations of multiarrangements

多元排列导数的积分表达式

• DOI: 10.48550/arxiv.2309.01287
• 发表时间: 2023
• 作者: Feigin M