Tangent Lifts of Poisson and Related Structures

Abstract

The derivation $d_T$ on the exterior algebra of forms on a manifold $M$ with values in the exterior algebra of forms on the tangent bundle $TM$ is extended to multivector fields. These tangent lifts are studied with applications to the theory of Poisson structures, their symplectic foliations, canonical vector fields and Poisson-Lie groups.

33 pages. To make the 1995-paper easily available

Cited by in corpus (49)

• Higher vector bundles and multi-graded symplectic manifolds
• Lie algebroids and Poisson-Nijenhuis structures
• Generalized Lie bialgebroids and Jacobi structures
• Algebroids - general differential calculi on vector bundles
• Jacobi structures revisited
• Graded contact manifolds and contact Courant algebroids
• Jacobi vector fields for Lagrangian systems on algebroids
• Linear and multiplicative 2-forms
• Nonholonomic Constraints: a New Viewpoint
• A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory
• Linear duals of graded bundles and higher analogues of (Lie) algebroids
• Brackets
• Higher order mechanics on graded bundles
• Modular classes of skew algebroid relations
• Pre-Courant Algebroids
• Multiplicative Dirac structures
• Time-dependent Mechanics and Lagrangian submanifolds of Dirac manifolds
• A geometric Hamilton--Jacobi theory for a Nambu--Poisson structure
• Tulczyjew triples: from statics to field theory
• Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids
• Geometry of quantum dynamics in infinite dimension
• VB-structures and generalizations
• Singular Improper Affine Spheres from a given Lagrangian Submanifold
• Polarisation of Graded Bundles
• Topological Membranes, Current Algebras and H-flux - R-flux Duality based on Courant Algebroids
• Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings
• Coadjoint orbits of Lie groupoids
• Isotropic submanifolds and the inverse problem for mechanical constrained systems
• Double vector bundles and duality
• Cauchy data space and multisymplectic formulation of conformal classical field theories
• Lie-Nijenhuis bialgebroids
• Reduction of pre-Hamiltonian actions
• In Search of Fundamental Discreteness in 2+1 Dimensional Quantum Gravity
• Lifting statistical structures
• Poisson-Nijenhuis groupoids
• On Filippov algebroids and multiplicative Nambu-Poisson structures
• Jacobi algebroids and Jacobi sigma models
• The ubiquity of the symplectic hamiltonian equations in mechanics
• Deformation of algebroid bracket of differential forms and Poisson manifold
• Complete and vertical lifts of Poisson vector fields and infinitesimal deformations of Poisson tensor
• A geometric Hamilton--Jacobi theory on a Nambu-Jacobi manifold
• Geometrical Mechanics on algebroids
• Jacobi fields of completely integrable Hamiltonian systems
• Lifts of Poisson structures to Weil bundles
• Jacobi structures on affine bundles
• Twisted Legendre transformation
• On characterization of Poisson and Jacobi structures
• A Darboux classification of homogeneous Pfaffian forms on graded manifolds
• The Tulczyjew triple in mechanics on a Lie group