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Canonical static concept record

Poisson manifold

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Summary

A smooth manifold whose smooth functions carry a compatible Poisson bracket.

Record metadata

Canonical name: Poisson manifold
Concept ID: poisson_manifold
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Differential geometry
Domains: Differential geometry, Algebra, Analysis
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

smooth manifold $M$

Data

Poisson bivector or bracket on smooth functions

Axioms / constraints

antisymmetry
Jacobi identity
Leibniz rule

Concept sources

Wikipedia — Poisson manifold — Poisson manifold · expository · source ID wp-poisson
nLab — Poisson manifold — Poisson manifold · mathematical reference · source ID nlab-poisson-manifold
Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Smooth manifold Poisson manifold

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: smooth_to_poisson
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: add Poisson bracket

Authored explanation

Add a Lie bracket on smooth functions that is a derivation in each argument.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Wikipedia — Poisson manifold — Poisson manifold · expository · source ID wp-poisson
Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds

Symplectic manifold Poisson manifold

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: symplectic_to_poisson
Relation type: Induced / forgotten induced-forgotten
Direction: source → target
Endpoint roles: source: Yields by induction / forgetting; target: Obtained by induction / forgetting from
Authored annotation: invert the symplectic form

Authored explanation

The inverse bivector of a symplectic form defines a Poisson bracket.

How to interpret this relation type

Pass canonically from a stronger object to structure it determines, or forget part of the data while retaining a valid weaker structure. The carrier may change under a canonical induced construction.

Relation sources

nLab — Symplectic manifold — Symplectic manifold · reference · source ID nlab-symplectic
Wikipedia — Poisson manifold — Poisson manifold · expository · source ID wp-poisson
Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.