Shifted double Poisson structures and noncommutative Poisson extensions

TL;DR

This paper develops a theory connecting shifted double Poisson structures on dg algebras with noncommutative Poisson extensions, cyclic homology, and derived moduli stacks, revealing their invariance and compatibility with Hamiltonian reduction.

Contribution

It introduces a comprehensive framework for noncommutative Poisson extensions and demonstrates their invariance and geometric implications in derived algebraic geometry.

Findings

Shifted double Poisson brackets induce graded Lie algebra structures on cyclic homology.

Noncommutative Poisson extensions are compatible with Hamiltonian reduction.

Shifted double Poisson structures are independent of cofibrant resolutions and induce structures on moduli stacks.

Abstract

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.