XXXVII Workshop on Geometric Methods in Physics 1-7.07.2018
VII School on Geometry and Physics 25-29.06.2018

Misha Feigin

Dunkl angular momenta algebra

Universal enveloping algebra of Lie algebra so(N) has a standard representation by differential operators in N-dimensional space, where generators are mapped to the operators of infinitesimal rotations. Elements of Dunkl angular momenta algebra are obtained by replacing partial derivatives with Dunkl operators associated with a Coxeter group W. The resulting algebra is a special subalgebra of the rational Cherednik algebra and it has good properties. Thus it is a PBW quadratic algebra, which may be thought of as a non-commutative deformation of the algebra of functions on the cone over Grassmanian of two-dimensional planes.

The central generator of the Dunkl angular momenta algebra can be identified with the non-local angular Calogero-Moser Hamiltonian associated with W. There is a version of Laplace-Runge-Lenz vector for the non-local Calogero-Moser Hamiltonian, which leads to an algebra isomorphic to the central quotient of Dunkl angular momenta algebra. The central quotient can also be obtained as the algebra of global sections of the sheaf of Cherednik algebras on the quadric of isotropic vectors.

The talk is based on joint works with Hakobyan, Nersessian and Thompson.

Event sponsored by:
of Bialystok
University of Bialystok

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