By Pavel Etingof
Calogero-Moser structures, which have been initially chanced on by means of experts in integrable platforms, are at present on the crossroads of many components of arithmetic and in the scope of pursuits of many mathematicians. extra in particular, those structures and their generalizations grew to become out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), illustration thought (double affine Hecke algebras, Lie teams, quantum groups), deformation concept (symplectic mirrored image algebras), homological algebra (Koszul algebras), Poisson geometry, and so forth. The aim of the current lecture notes is to offer an creation to the idea of Calogero-Moser structures, highlighting their interaction with those fields. on account that those lectures are designed for non-experts, the writer supplies brief introductions to every of the topics concerned and gives a couple of workouts. A book of the ecu Mathematical Society (EMS). disbursed in the Americas through the yankee Mathematical Society.
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Extra resources for Calogero-Moser Systems and Representation Theory (Zurich Lectrues in Advanced Mathematics)
T 1 x, p ! tp, polynomial in p of degree m. From these properties of S it is clear that S is not a polynomial (its degree in x is m d < 0). On the other hand, since Y commutes with h , the Poisson bracket of S with p 2 is zero. 5 follows from the following lemma. 7. x; p/ 7! x; p/ be a rational function on h ˚ h , which is polynomial in p 2 h . Let f W h ! v/ is identically zero in p . for example, f D p 2 /. Suppose that the Poisson bracket ff; Sg is equal to zero. Then S is a polynomial: S 2 CŒh ˚ h .
Namely, let us first calculate L02 . x/. x/ 1 h˛ C ; where h˛ D Œe˛ ; f˛ . k C 1/ (this is obtained by a direct computation using that e˛ D xi @j ; f˛ D xj @i ). 6. k C 1/=x 2 . 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.
The proof is obtained immediately by considering the bar resolution of the bimodule A: ! A ˝ A ˝ A ! A ˝ A ! a1 ˝ a2 ˝ and the map @n W A˝n ! a1 ˝ a2 ˝ ˝ an /c D ba1 ˝ a2 ˝ 1 ˝ an c; is given by the formula ˝ an / D a1 a2 ˝ ˝ an C . A; N / ! A; M ˝A N /; induced by tensoring of cochains. A/. A/ is the quotient of the Lie algebra of derivations of A by inner derivations. A/; it can be shown that this algebra is supercommutative. 3 Hochschild cohomology and deformations Let A0 be an algebra, and let us look for 1-parameter deformations A D A0 ŒŒ„ of A0 .
Calogero-Moser Systems and Representation Theory (Zurich Lectrues in Advanced Mathematics) by Pavel Etingof