By Aldo Andreotti, Wilhelm Stoll
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Additional resources for Analytic and Algebraic Dependence of Meromorphic Functions
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 .
Analytic and Algebraic Dependence of Meromorphic Functions by Aldo Andreotti, Wilhelm Stoll