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Quasi-invariants and quantum integrals of the deformed Calogero-Moser systems. (English) Zbl 1034.81024
The authors of this interesting paper study the algebra of quantum integrals for two series $$\mathcal A$$$$_n(m)$$ and $$\mathcal C$$$$_{n+1}(m,l)$$ of the deformed quantum Calogero-Moser (CM) systems discovered by O. A. Chalykh, M. V. Feigin and A. P. Veselov (1998). It turns out that if all the parameters are integers then these series are the only non-Coxeter cases among the deformed CM systems. The corresponding operators $$L$$ can be expressed by Laplace operator $$\triangle$$ in $$(n+1)$$-dimensional Euclidean space and some summands which depend on parameters $$m,k,l$$ under the relation $$k=(2m+1)/(2l+1)$$. The Calogero operator can be obtained when $$m=1$$. The deformation of the generalized CM operator related to the root system $$\mathcal C$$$$_{n+1}$$ is expressed by a differential operator corresponding to the case $$m=l$$. Here the authors are interested in the case when all the parameters (multiplicities) in these operators are integers. The main result is that the ring of quasi-invariants $$Q^{\mathcal{A}}$$ related to the deformed configurations $$\mathcal A$$$$_n(m)$$ and $$\mathcal C$$$$_{n+1}(m,l)$$ is a free module over the subring of invariant polynomials. A certain polynomial subalgebra $$\mathcal P$$$$^{\mathcal{A}}\subset Q^{\mathcal{A}}$$ is introduced, and it is shown that $$Q^{\mathcal{A}}$$ is free as a module over $$\mathcal P$$$$^{\mathcal{A}}$$. In the two-dimensional case the authors find the explicit formulas for the Poincaré series of the corresponding rings of quasi-invariants. Some deformed $$m$$-harmonic polynomials are defined and computed explicitly in the $$\mathcal A$$$$_2(m)$$ case.

##### MSC:
 81R12 Groups and algebras in quantum theory and relations with integrable systems 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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