Calogero-Moser Systems and Representation TheoryCalogero-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. |
Contents
Introduction | 1 |
Quantum mechanics quantum integrable systems and | 5 |
Deformation theory | 21 |
Moment maps Hamiltonian reduction and the LevasseurStafford | 29 |
CalogeroMoser systems associated to finite Coxeter groups | 47 |
1 | 53 |
Symplectic reflection algebras | 59 |
Deformationtheoretic interpretation of symplectic reflection algebras | 65 |
Common terms and phrases
action of G action-angle variables algebraic deformation algebraic variety associative algebra Azumaya algebra bivector Bo,c Bt,c Calogero Calogero-Moser space Calogero-Moser system classical mechanics coadjoint orbit Cohen-Macaulay coordinates corresponding D(hreg defined Definition deformation quantization deformation theory degree denote differential operators Dunkl operators eigenvalue elements example Exercise filtration finite dimensional formal deformation G-invariant H₁ H₂ Hamilton's equations Harish-Chandra Hochschild cohomology homological dimension homomorphism Ht,c Ht,ce ideal irreducible representation isomorphism Kontsevich Koszul Lecture lemma Let g Lie algebra Lie group linear linearly independent map µ Math matrix Mc(t module moment map nondegenerate Olshanetsky-Perelomov open set phase space Poisson algebra Poisson bracket Poisson manifold Poisson structure polynomial Proof Proposition quantum Hamiltonian reduction quantum integrable system quantum moment map quotient rational Cherednik algebra spherical subalgebra symmetry symplectic form symplectic manifold symplectic reflection algebras Theorem universal deformation Weyl Weyl(V zero