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