Reduction along strong Dirac maps

TL;DR

This paper introduces a comprehensive reduction procedure along strong Dirac maps, generalizing Poisson momentum maps, and recovers many known structures while presenting new examples in Poisson, quasi-Poisson, and Dirac geometry.

Contribution

It develops a unifying framework for reduction along strong Dirac maps, extending classical Poisson and quasi-Poisson reduction methods and introducing novel structures.

Findings

Recovered classical Poisson and quasi-Poisson structures

Introduced new examples of reduced Poisson, quasi-Poisson, and Dirac structures

Established quasi-Poisson analogues of spaces in geometric representation theory

Abstract

We develop a general procedure for reduction along strong Dirac maps, which are a broad generalization of Poisson momentum maps. We recover a large number of familiar constructions in Poisson and quasi-Poisson geometry, and we introduce new examples of Poisson, quasi-Poisson, and Dirac reduced structures. In particular, we obtain quasi-Poisson analogues of several classes of spaces that are studied in geometric representation theory.