Angular Cherednik Algebras and Integrability

可积系统描述了粒子之间的相互作用，其中可以获得大量关于粒子行为的精确信息。这既与经典力学系统有关，在经典力学系统中，人们对轨迹和守恒量感兴趣，也与量子系统有关，在量子系统中，守恒量最终有助于确定光谱。这种情况很少见，而且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粒子行为的附加属性。因此，可积系统往往与代数和几何有很深的联系，这些联系在过去已经被非常卓有成效地探索过了。例如，著名的Calogero-Moser系统描述了直线上两两相互作用的粒子，其势能与粒子之间的平方距离成反比；它与对称空间的几何、代数几何以及在过去20年中蓬勃发展的切莱德尼克代数密切相关。这个跨学科项目的目标是将可积系统的专业知识与几何表示理论的专业知识结合在一起，以发现新的可积系统和相关的代数结构，并进一步与奇点几何和李理论建立有趣的联系。该项目的一个关键对象是一个角度版本的Calogero-Moser系统，它对应于更高维度球体上的运动。在这种情况下出现的非交换代数由于其新颖性和更大的复杂性，人们对它的理解很少，研究的也更少。我们将发展这些代数的表示理论。在几何上，这些代数量子化了一类新的辛奇点。我们将使用几何和表示论方法来研究这些奇异空间。我们希望这将导致一类美丽的例子，结合辛商奇点和幂零轨道闭包，这是几何和代数之间相互作用的一个引人注目的新例证。我们还将发现并研究Calogero-Moser系统的相对论扩展的角度版本，这些扩展有望成为新的可积系统。相应的代数结构将被揭示：它们预计将涉及切莱德尼克代数和量子群的新混合，而量子群在许多数学领域是无处不在的。

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.