Derived Poisson Structures

We introduce the notion of a derived Poisson structure on an associative algebra A. This structure is characterized by the property of being the weakest structure on A that induces natural (graded) Poisson structures on the derived moduli spaces of finite-dimensional representations of A. A derived Poisson structure on A gives rise to a graded (super) Lie algebra structure on the full cyclic homology HC(A) and can be viewed as a higher homological extension of noncommutative Poisson structures in the sense of M. Kontsevich.