Banach Lie algebroids and Dirac structures
نویسنده
چکیده
We consider the category of anchored Banach vector bundles and we discuss the notion of semispray. Adding on the set of sections of an anchored Banach vector bundle a Lie bracket with some properties one gets the notion of Lie algebroid. We prove that the Lie algebroids form also a category. A Dirac structure on a Banach manifold M is defined as a subbundle of the big tangent bundle TM ⊕ T ∗M that equals its orthocomplement with respect to the standard neutral metric and is closed with respect to the Courant bracket. Various characterizations of this closeness are provided. We show that with a convenient anchor any Dirac structure becomes a Banach Lie algebroid. Some examples are included. M.S.C. 2010: 53D17, 58A99.
منابع مشابه
Stability of higher order singular points of Poisson manifolds and Lie algebroids
We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algeb...
متن کاملQuasi-Poisson structures as Dirac structures
We show that quasi-Poisson structures can be identified with Dirac structures in suitable Courant algebroids. This provides a geometric way to construct Lie algebroids associated with quasi-Poisson spaces.
متن کاملClifford Algebroids and Nonholonomic Einstein–Dirac Structures
We propose a new framework for constructing geometric and physical models on spacetimes provided with Lie algebroid symmetry, i.e. manifolds provided with additional anchor and generalized Lie algebra commutator structures. The approach is related to the geometry of moving nonholonomic frames with associated nonlinear connections. A strict application of such geometric methods to spinor fields ...
متن کاملThe graded Jacobi algebras and (co)homology
Jacobi algebroids (i.e. a ‘Jacobi versions’ of Lie algebroids) are studied in the context of graded Jacobi brackets on graded commutative algebras. This unifies varios concepts of graded Lie structures in geometry and physics. A method of describing such structures by classical Lie algebroids via certain gauging (in the spirit of E.Witten’s gauging of exterior derivative) is developed. One cons...
متن کاملReduction of Jacobi Manifolds via Dirac Structures Theory
We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold (M,Λ, E) for which 1 is an admissible function and Jacobi quotient manifolds of M . We study Jacobi reductions ...
متن کامل