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Автоморфные формы и их приложения
23 мая 2017 г. 18:00–20:00, г. Москва, ул. Усачева 6, аудитория 306

Universal Dunkl operators: Algebra, Combinatorics, Graph Theory, Integrable Systems and LDT

А. Н. Кириллов

Research Institute for Mathematical Sciences, Kyoto University

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Аннотация: I introduce certain noncommutative (inhomogeneous) quadratic algebra together with distinguished set of elements, called the (additive) Dunkl elements. The basic property of Dunkl elements is that they generate a commutative subalgebra inside of the noncommutative quadratic algebra we have introduced.
The main objective of our research concerning the quadratic algebra under consideration is to identify the commutative subalgebra generated by (additive) Dunkl elements with some well-known commutative algebras, such as Classical and Quantum Cohomology of certain varieties, algebras generated by integrals of motion of certain Integrable Systems, algebras associated with hyperplane arrangements, algebras associated with Low Dimensional Topology (LDT), and some others.
The main step to describe such an identification is to find a representation of the (noncommutative) algebra we have defined, which can be used for our purposes.
We are planning to talk about some common features and applications of the items below. I will try to touch only the key points (in my opinion) of that items.
1. The Arnold type representations. These types of representations are useful for application to Combinatorics and Graph Theory;
2. The Bruhat type representations. These types of representations can be used for applications to Classical and Quantum Schubert and Grothendieck Calculi of (type A) flag varieties;
3. The Kohno-Drinfeld type representations. Conjecturally these types of representations are related with LDT of virtual knots and links;
4. The Calogero-Moser type representations (including rational, trigonometric, elliptic and higher genus versions). We touch a problem of lifting some well-known elliptic identities to noncommutative setup. We will point out some connections with Integrable Systems.

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