Shifted double Poisson structures and noncommutative Poisson extensions

We develop a theory of noncommutative Poisson extensions. For an augmented dg algebra $A$, we show that any shifted double Poisson bracket on $A$ induces a graded Lie algebra structure on the reduced cyclic homology. Under the Kontsevich--Rosenberg principle, we further prove that the noncommutative Poisson extension is compatible with noncommutative Hamiltonian reduction. Moreover, we show that shifted double Poisson structures are independent of the choice of cofibrant resolutions and that they induce shifted Poisson structures on the derived moduli stack of representations.