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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.

We solve some noncommutative analogue of the Noether's problem for the reflection groups by showing that the skew field of fractions of the invariant subalgebra of the Weyl algebra under the action of any finite complex reflection group is a Weyl field, that is isomorphic to the skew field of fractions of some Weyl algebra. We also extend this result to the invariants of the ring of differential operators on any finite dimensional torus. The results are applied to obtain analogs of the Gelfand-Kirillov Conjecture for Cherednik algebras and Galois algebras.

Let M be a smooth, simply-connected, closed oriented manifold, and L M the free loop space of M. Using a Poincaré duality model for M, we show that the reduced equivariant homology of L M has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.

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