Shifted symplectic and Poisson structures on global quotients

TL;DR

This paper characterizes shifted symplectic and Poisson structures on global quotients of derived stacks using Cartan-de Rham complexes, extending the theory to non-reductive groups and providing a Cartan model for polyvector fields.

Contribution

It introduces a novel characterization of shifted symplectic and Poisson structures via Cartan complexes for both reductive and non-reductive group quotients.

Findings

Characterization of shifted symplectic structures using Cartan-de Rham complex.

Extension of the characterization to non-reductive groups with Getzler's complex.

Construction of a Cartan model for polyvector fields on global quotients.

Abstract

For a derived stack obtained as a quotient of a derived affine scheme by a reductive group, we show that shifted symplectic structures can be characterized by the Cartan-de Rham complex. For non-reductive groups, we also show the analogous statement for Getzler's extension of the Cartan-de Rham complex. Dually, we construct a Cartan model for polyvector fields on global quotients by reductive groups, and show that shifted Poisson structures can be characterized by it.