Angular Cherednik Algebras and 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.