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Generalized Calogero-Moser systems from rational Cherednik algebras. (English) Zbl 1243.81089
A Calogero-Moser (CM) system is defined by a quantum integrable Hamiltonian describing the interaction of \(n\) particles on the line with the inverse square potential. These systems are related to the root systems of finite Coxeter groups and the Hamiltonians can be greatly generalized by studying the representations of the rational Cherednik algebra associated with each root system. In this paper the author considers ideals of polynomials vanishing on the \(W\)-orbits of the intersections of mirrors of a finite reflection group \(W\). He determines all such ideals invariant under the corresponding rational Cherednik algebra, so that they form submodules in the polynomial module. He shows that for every such ideal of a real reflection group a quantum integrable system can be defined. Carrying out this procedure for the classical Coxeter groups he finds and classifies all corresponding CM systems, some of them new. He shows that a quadratic term can be added to each Hamiltonian such that it remains integrable. He also extends his method to classify all generalized CM systems associated with the exceptional Coxeter groups, providing a table of the results. Finally, he studies polynomial ideals for finite complex reflection groups. The paper is highly technical but the background and development are clearly explained.

MSC:
81R12 Groups and algebras in quantum theory and relations with integrable systems
70E40 Integrable cases of motion in rigid body dynamics
16G99 Representation theory of associative rings and algebras
70F10 \(n\)-body problems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
17B22 Root systems
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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