Abstract
Let 𝔤 be a simple Lie algebra of rank l, 𝔥 a Cartan subalgebra and W the corresponding Weyl group. The space 𝔥⊕𝔥* is naturally equipped with a symplectic structure and the invariant algebra S(𝔥⊕𝔥*) W inherits a Poisson structure which can be deformed by the invariant algebra where A l denotes the Weyl algebra of rank l. The purpose of this article is to compute the Poisson homology group of degree zero of S(𝔥⊕𝔥*) W and to compare it to the Hochschild homology group of degree zero of . These two groups have the same dimensions in rank 2 which are equal respectively to 1.2 and 3 in the types A 2, B 2 and G 2.
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