By Pavel Etingof
Calogero-Moser platforms, which have been initially found through experts in integrable structures, are presently on the crossroads of many parts of arithmetic and in the scope of pursuits of many mathematicians. extra particularly, those structures and their generalizations grew to become out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), illustration concept (double affine Hecke algebras, Lie teams, quantum groups), deformation thought (symplectic mirrored image algebras), homological algebra (Koszul algebras), Poisson geometry, and so on. The objective of the current lecture notes is to offer an advent to the speculation of Calogero-Moser platforms, highlighting their interaction with those fields. because those lectures are designed for non-experts, the writer supplies brief introductions to every of the topics concerned and gives a few workouts. A ebook of the ecu Mathematical Society (EMS). disbursed in the Americas via the yank Mathematical Society.
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Extra resources for Calogero-Moser Systems and Representation Theory (Zurich Lectrues in Advanced Mathematics)
The usual quantum reduction described above is a special case of this, when I is the augmentation ideal. The following example is a quantization of the Kazhdan–Kostant–Sternberg construction of the Calogero–Moser space given in Section 1. 14. g/ as above. x1 ; : : : ; xn /, where f is a Laurent polynomial of degree 0. We regard Wk as a g-module by pulling it back to g under the natural projection g ! sln . g/. A; g; Ik /. g/g . g/g ! k/ be the kernel of HC k . 15 (Etingof, Ginzburg, [EG]). h/W , HC 0 D HC.
It is unique up to an isomorphism (which may involve an automorphism of kŒŒ„). A0 / D 0, deformation theory allows us to completely classify 1-parameter flat formal deformations of A0 . In particular, we see that the “moduli space” parametrizing formal deformations of A0 is a smooth space – it is the formal neighborhood of zero in H . A0 / is nonzero then in general the universal deformation parametrized by H does not exist, as there are obstructions to deformations. In this case, the moduli space of deformations will be a closed subscheme of H , which is often singular.
Let us compute it more explicitly on the open set Un using the coordinates xi ; pi . xi D xi pi : xi xj : xj /2 Let us introduce “additive” coordinates xi D log xi , pi D xi pi . It is easy to check that these coordinates are canonical. x=2/. Remark. x=2/. 9 Notes 1. For generalities on classical mechanics, symmetries of a mechanical system, reduction of order using symmetries, integrable systems, action-angle variables we refer the reader to [Ar]. Classical Calogero–Moser systems go back to the papers [Ca], [Mo]; their construction using reduction along orbit is due to Kazhdan, Kostant, and Sternberg, [KKS].
Calogero-Moser Systems and Representation Theory (Zurich Lectrues in Advanced Mathematics) by Pavel Etingof