Poisson bivectors and Poisson brackets on affine derived stacks

Let A be a commutative dg algebra concentrated in degrees (-infinity, m], and let Spec A be the associated derived stack. We give two proofs of the existence of a canonical map from the moduli space of shifted Poisson structures (in the sense of [16]) on Spec A to the moduli space of homotopy (shifted) Poisson algebra structures on A. The first makes use of a more general description of the Poisson operad and of its cofibrant models, while the second is more computational and involves an explicit resolution of the Poisson operad. (C) 2015 Elsevier Inc. All rights reserved.