Noncommutative derived Poisson reduction

TL;DR

This paper develops a noncommutative derived Poisson reduction framework using double Poisson structures, connecting noncommutative and commutative geometries through representation functors and categorical insights.

Contribution

It introduces a novel noncommutative derived Poisson reduction procedure based on double Poisson structures and explores categorical properties leading to new notions like noncommutative group schemes.

Findings

Recovery of commutative constructions via representation functor

Introduction of noncommutative group schemes and actions

Establishment of a broader noncommutative-commutative correspondence

Abstract

In this paper we propose a procedure for a noncommutative derived Poisson reduction, in the spirit of the Kontsevich-Rosenberg principle: "a noncommutative structure of some kind on $A$ should give an analogous commutative structure on all schemes ${\mathrm{Rep}}_n(A)$". We use double Poisson structures as noncommutative Poisson structures and noncommutative Hamiltonian spaces -- as first introduced by M. Van den Bergh -- to define (derived) zero loci of Hamiltonian actions and a noncommutative Chevalley-Eilenberg and BRST constructions, showing how we recover the corresponding commutative constructions using the representation functor. In a dedicated final short section we highlight how the categorical properties of the representation functor lead to the natural introduction of new interesting notions, such as noncommutative group schemes, group actions, or Poisson-group schemes, which could help to understand the previous results in a different light, and in future research generalise them into a broader, clearer correspondence between noncommutative and commutative equivariant geometry.