Abstract. Let A be a Koszul (or more generally, N -Koszul) Calabi-Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on A, which induces a graded Lie algebra structure on the cyclic homology of A; moreover, we show that the Hochschild homology of A is a Lie module over the cyclic homology and the Connes long exact sequence is in fact a sequence of Lie modules. Finally, we show that the Leibniz-Loday bracket associated to the derived non-commutative Poisson structure on A is naturally mapped to the Gerstenhaber bracket on the Hochschild cohomology of its Koszul dual algebra and hence on that of A itself. Relations with some other brackets in literature are also discussed and several examples are given in detail.

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Let G be the group of unimodular automorphisms of a free associative Calgebra on two generators. A theorem of G. Wilson and the first author [BW] asserts that the natural action of G on the Calogero-Moser spaces Cn is transitive for all n ∈ N. We extend this result in two ways: first, we prove that the action of G on Cn is doubly transitive, meaning that G acts transitively on the configuration space of ordered pairs of distinct points in Cn; second, we prove that the diagonal action of G on Cn 1 × Cn 2 × · · · × Cn m is transitive provided n 1 , n 2 , . . . , nm are pairwise distinct numbers.

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