Calogero-Moser systems, which were originally discovered by specialists in integrable systems, are currently at the crossroads of many areas of mathematics and within the scope of interests of many mathematicians. More specifically, these systems and their generalizations turned out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), representation theory (double affine Hecke algebras, Lie groups, quantum groups), deformation theory (symplectic reflection algebras), homological algebra (Koszul algebras), Poisson geometry, etc. The goal of the present lecture notes is to give an introduction to the theory of Calogero-Moser systems, highlighting their interplay with these fields. Since these lectures are designed for non-experts, the author gives short introductions to each of the subjects involved and provides a number of exercises. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.
Author(s): Pavel Etingof
Publisher: American Mathematical Society
Year: 2007
Language: English
Commentary: no
Pages: 104
Cover......Page 1
Zurich Lectures in Advanced Mathematics......Page 3
Title......Page 4
ISBN 978-3-03719-034-0......Page 5
Dedication......Page 6
Contents......Page 8
Introduction......Page 12
Poisson manifolds......Page 16
Moment maps......Page 17
Hamiltonian reduction......Page 18
Calogero–Moser space......Page 20
Notes......Page 21
Classical mechanics......Page 22
Symmetries in classical mechanics......Page 23
Integrable systems......Page 24
Action-angle variables of integrable systems......Page 25
The Calogero–Moser system......Page 26
Coordinates on C_n and the explicit form of the Calogero–Moser system......Page 27
The trigonometric Calogero–Moser system......Page 29
Notes......Page 30
Hochschild cohomology......Page 32
Universal deformation......Page 34
Quantization of Poisson algebras and manifolds......Page 35
Algebraic deformations......Page 37
Notes......Page 38
Quantum moment maps and quantum Hamiltonian reduction......Page 40
The Levasseur–Stafford theorem......Page 41
Hamiltonian reduction with respect to an ideal in U(g)......Page 46
Quantum reduction in the deformational setting......Page 47
Notes......Page 48
Quantum mechanics......Page 50
Quantum integrable systems......Page 51
Constructing quantum integrable systems by quantum Hamiltonian reduction......Page 53
The quantum Calogero–Moser system......Page 54
Notes......Page 55
Dunkl operators......Page 58
Olshanetsky–Perelomov operators......Page 59
Classical Dunkl operators and Olshanetsky–Perelomov Hamiltonians......Page 61
Notes......Page 63
Rational Cherednik algebra and the Poincaré–Birkhoff–Witt theorem......Page 64
The localization lemma and the basic properties of M_c......Page 66
The SL_2-action on H_{t,c}......Page 67
Notes......Page 68
The PBW theorem for symplectic reflection algebras......Page 70
Koszul algebras......Page 71
Proof of Theorem 8.3......Page 72
The spherical subalgebra of the symplectic reflection algebra......Page 73
Notes......Page 74
Hochschild cohomology of semidirect products......Page 76
Notes......Page 78
The module H_{t,c}e......Page 80
Finite dimensional representations of H_{0,c}......Page 81
Azumaya algebras......Page 82
Cohen–Macaulay property and homological dimension......Page 83
Proof of Theorem 10.10......Page 85
The space M_c for G=S_n......Page 86
Generalizations......Page 88
Notes......Page 89
Verma and irreducible lowest weight modules over H_1,c......Page 90
Category Ø......Page 92
The Frobenius property......Page 93
Representations of the rational Cherednik algebra of type A......Page 94
The results.......Page 95
Proof of Theorem 11.16.......Page 96
Notes......Page 97
Bibliography......Page 98
Index......Page 102
Back Cover......Page 104
