Concurring reduction schemes for Dirac structures

TL;DR

This paper investigates conditions under which Dirac structures maintain their concurrence relation during reduction, extending classical results and providing new procedures for constructing common witnesses.

Contribution

It characterizes minimal Dirac reduction schemes and proves that concurring Dirac structures have concurring reductions if they share a witness, extending the Marsden-Ratiu theorem.

Findings

Two procedures for constructing common witnesses are introduced.

Concurring Dirac structures have concurring reductions when sharing a witness.

Examples include Hamiltonian actions and Dirac-Nijenhuis manifolds.

Abstract

The notion of \emph{concurrence} was recently proposed as the natural compatibility relation between Dirac structures, generalizing the commutativity of two Poisson structures. We address the question of when a reduction scheme -- that is, a way to induce a Dirac structure on a quotient of a submanifold -- respects this relation. After characterizing the minimal scheme of \emph{Dirac reduction}, we prove that two concurring Dirac structures have concurring reductions whenever they share a common \emph{witness}, extending to Dirac geometry the reduction of the Marsden-Ra\cb{t}iu theorem. Two procedures for constructing such common witnesses are given, the second being the Dirac counterpart of Magri's original recipe in bihamiltonian geometry. Examples drawn from Hamiltonian actions, Dirac-Nijenhuis manifolds, and complex Dirac structures conclude the paper and illustrate our methods.