Concurring reduction schemes for Dirac structures
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918472927150080 |
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| author | Aguero, Dan Arsie, Alessandro Frejlich, Pedro Mencattini, Igor |
| author_facet | Aguero, Dan Arsie, Alessandro Frejlich, Pedro Mencattini, Igor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26196 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Concurring reduction schemes for Dirac structures Aguero, Dan Arsie, Alessandro Frejlich, Pedro Mencattini, Igor Symplectic Geometry 53D17 (Primary) 53D18, 53D20 (Secondary) 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. |
| title | Concurring reduction schemes for Dirac structures |
| topic | Symplectic Geometry 53D17 (Primary) 53D18, 53D20 (Secondary) |
| url | https://arxiv.org/abs/2604.26196 |