By Pavel Etingof

Calogero-Moser structures, which have been initially found by way of experts in integrable platforms, are at present on the crossroads of many parts of arithmetic and in the scope of pursuits of many mathematicians. extra in particular, those platforms and their generalizations became out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), illustration idea (double affine Hecke algebras, Lie teams, quantum groups), deformation thought (symplectic mirrored image algebras), homological algebra (Koszul algebras), Poisson geometry, and so forth. The aim of the current lecture notes is to provide an advent to the speculation of Calogero-Moser structures, highlighting their interaction with those fields. considering that those lectures are designed for non-experts, the writer offers brief introductions to every of the topics concerned and offers a few workouts. A book of the eu Mathematical Society (EMS). disbursed in the Americas through the yank Mathematical Society.

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**Example text**

10. B/ exists. M; C/, and t1 ; : : : ; tr are the coordinates on H corresponding to this basis. 6 Algebraic deformations Formal deformations of algebras often arise from algebraic deformations. The most naive definition of an algebraic deformation is as follows. k/. Let I0 be the maximal ideal corresponding to 0. 11. An algebraic deformation over B ´ kŒ† of an algebra A0 is a Balgebra A which is a free B-module, together with the identification Á0 W A=I0 A ! A0 of the zero-fiber of A with A0 as algebras.

G„ /, where g„ is the Lie algebra over CŒ„ which is equal to gŒ„ as a vector space, with bracket Œa; b„ ´ „Œa; b. 3. Quantum torus. xy D qyx/, † D k . 7 Notes 1. For generalities on Hochschild cohomology, see the book [Lo]. The basics of deformation theory of algebras are due to Gerstenhaber [Ge2]. 2. The notion of deformation quantization was proposed in the classical paper [BFFLS]; in this paper the authors ask the question whether every Poisson manifold admits a quantization, which was solved positively by Kontsevich.

The question whether there are other obstructions. The answer to this question is yes and no. Namely, if you do not pick 2 carefully, you may be unable to find 3 , but you can always pick 2 so that 3 exists, and there is a similar situation in higher orders. 8 (Kontsevich, [Ko1], [Ko2]). Any Poisson structure on A0 can be quantized. Moreover, there is a natural bijection between products up to an isomorphism equal to 1 modulo „, and Poisson brackets 0 C „ 1 C „2 2 C up to a formal diffeomorphism equal to 1 modulo „, such that the quasiclassical limit of is 0 .