By Pavel Etingof
Calogero-Moser platforms, which have been initially found through experts in integrable structures, are at the moment on the crossroads of many components of arithmetic and in the scope of pursuits of many mathematicians. extra particularly, those platforms and their generalizations became out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), illustration conception (double affine Hecke algebras, Lie teams, quantum groups), deformation concept (symplectic mirrored image algebras), homological algebra (Koszul algebras), Poisson geometry, and so on. The aim of the current lecture notes is to offer an advent to the idea of Calogero-Moser structures, highlighting their interaction with those fields. considering that those lectures are designed for non-experts, the writer supplies brief introductions to every of the themes concerned and offers a couple of workouts. A book of the ecu Mathematical Society (EMS). dispensed in the Americas through the yankee Mathematical Society.
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Extra resources for Calogero-Moser Systems and Representation Theory
Hreg //W ! hreg /W . 9 (Heckman, [Hec]). Let fy1 ; : : : yr g be an orthonormal basis of h. Then we have r Á X x Dy2i D L: m iD1 Proof. h/W ! h/ corresponding to B. Dy @y /. ˛s ; ˛s /. Sh/W is free. S h/W . 2 x defines a quantum integrable system. 10. Dy1 ; : : : ; Dyr // are commuting quantum integrals of xDL x 1. L 50 6 Calogero–Moser systems associated to finite Coxeter groups Proof. b2 /. To derive from this the quantum integrability of the operator L, we will prove the following proposition.
Let us denote the algebra defined in the proposition by Hc0 . Then according to the results of the previous section, we have a surjective homomorphism W Hc0 ! c; „/. Let us show that this homomorphism is injective. For this purpose assume that yi is a basis of h, and xi is the dual basis of h . c; „/mi xi i g iD1 iD1 are linearly independent. But this follows from the obvious fact that the symbols of these elements in CW Ë CŒh hreg Œ„ are linearly independent. The proposition is proved. 54 7 The rational Cherednik algebra It is more convenient to work with algebras defined by generators and relations than with subalgebras of a given algebra generated by a given set of elements.
Remark. Note that unlike the classical case, in the quantum case the coefficient in front of the potential is an essential parameter and cannot be removed by rescaling. 5 Notes 1. Quantum integrable systems have been studied for more than twenty years; let us mention, for instance, the paper [OP], which is relevant to the subject of these lectures. g. [He]. The quantum Calogero–Moser system (more precisely, its trigonometric deformation) appeared in [Su]. 2. 5 Notes 45 for instance, in [EG]. xi xj / is also integrable and may be obtained by performing reduction from T G (rather than T g), both classically and quantum mechanically, mimicking the Kazhdan–Kostant– Sternberg construction.
Calogero-Moser Systems and Representation Theory by Pavel Etingof