  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.
