Description of coupling in the category of transitive Lie algebroids (original) (raw)
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The existence of coupling in the category of transitive lie algebroid
The coupling of the tangent bundle T M with the Lie algebra bundle L ([5], Definition 7.2.2) plays the crucial role in the classification of the transitive Lie algebroids for Lie algebra bundle L with fixed finite dimensional Lie algebra g as a fiber of L. Here we give a necessary and sufficient condition for the existence of such a coupling. Namely we define a new topology on the group Aut (g) of all automorphisms of Lie algebra g and show that tangent bundle T M can be coupled with the Lie algebra bundle L if and only if the Lie algebra bundle L admits a local trivial structure with structural group endowed with such new topology.
Algebraic Constructions in the Category of Lie Algebroids
2011
A generalized notion of a Lie algebroid is presented. Using this, the Lie algebroid generalized tangent bundle is obtained. A new point of view over (linear) connections theory on a fiber bundle is presented. These connections are characterized by o horizontal distribution of the Lie algebroid generalized tangent bundle. Some basic properties of these generalized connections are investigated. Special attention to the class of linear connections is paid. The recently studied Lie algebroids connections can be recovered as special cases within this more general framework. In particular, all results are similar with the classical results. Formulas of Ricci and Bianchi type and linear connections of Levi-Civita type are presented.
Russian Journal of Mathematical Physics, 2015
Transitive Lie algebroids have specific properties that allow to look at the transitive Lie algebroid as an element of the object of a homotopy functor. Roughly speaking each transitive Lie algebroids can be described as a vector bundle over the tangent bundle of the manifold which is endowed with additional structures. Therefore transitive Lie algebroids admits a construction of inverse image generated by a smooth mapping of smooth manifolds. Due to to K.Mackenzie ([1]) the construction can be managed as a homotopy functor T LAg from category of smooth manifolds to the transitive Lie algebroids. The functor T LAg associates with each smooth manifold M the set T LAg(M) of all transitive algebroids with fixed structural finite dimensional Lie algebra g. Hence one can construct ([4],[5]) a classifying space Bg such that the family of all transitive Lie algebroids with fixed Lie algebra g over the manifold M has one-to-one correspondence with the family of homotopy classes of continuous maps [M, Bg]: T LAg(M) ≈ [M, Bg]. It allows to describe characteristic classes of transitive Lie algebroids from the point of view a natural transformation of functors similar to the classical abstract characteristic classes for vector bundles and to compare them with that derived from the Chern-Weil homomorphism by J.Kubarski([3]). As a matter of fact we show that the Chern-Weil homomorphism does not cover all characteristic classes from categorical point of view.
The generalized Lie algebroids and their applications
2010
In this paper we introduce the notion of generalized Lie algebroid and we develop a new formalism necessary to obtain a new solution for the Weistein's Problem [53]. Many applications emphasize the importance and the utility of this new framework determined by the introduction of generalized Lie algebroids. We introduce and develop the exterior differential calculus for generalized Lie algebroids and, in this general framework, we establish the structure equations of Maurer-Cartan type. In particular, we obtain a new point of view over the exterior differential calculus for Lie algebroids. Using the (generalized) Lie algebroids theory, we build the Lie algebroid generalized tangent bundle and, using that, we obtain a new method by determining the (linear) connections for fiber bundles, in general, and for vector bundles, in particular. Using the linear connections theory we develop the study of the geometry of vector bundles. Moreover, using the connections theory, we develop the geometry of total space of the generalized tangent bundle for a vector bundle. We present a geometric description of metrizability for the total space of the Lie algebroid generalized tangent bundle, where we extend the notions of generalized Lagrange space, Lagrange space and Finsler space. Using the Lie algebroid generalized tangent bundle of a generalized Lie algebroid, we introduce and develop a mechanical systems theory and we present a Lagrangian formalism for these mechanical systems. In particular, using the Lie algebroid generalized tangent bundle of a Lie algebroid, we obtain a new solution for the Weinstein's Problem. A geometric description of metrizability for the total space of the Lie algebroid generalized tangent bundle for dual vector bundle is presented. We extend the notions of generalized Hamilton space, Hamilton space and Cartan space. Using the Lie algebroid generalized tangent bundle of dual of a generalized Lie algebroid, we introduce and develop the dual mechanical systems theory and we present a Hamiltonian formalism for dual mechanical systems. Finally, we introduce and develop the concept of (horizontal) Legendre equivalence between a vector bundle and its dual vector bundle. We remark that, if the morphisms used are identities morphisms, then we obtain similar results to the classical results, but which are not classical results though.
The Cohomology of Transitive Lie Algebroids
2007
For a transitive Lie algebroid A on a connected manifold M and its a representation on a vector bundle F, we study the localization map Y^1: H^1(A,F)-> H^1(L_x,F_x), where L_x is the adjoint algebra at x in M. The main result in this paper is that: Ker Y^1_x=Ker(p^{1*})=H^1_{deR}(M,F_0). Here p^{1*} is the lift of H^1(\huaA,F) to its counterpart over the universal covering space of M and H^1_{deR}(M,F_0) is the F_0=H^0(L,F)-coefficient deRham cohomology. We apply these results to study the associated vector bundles to principal fiber bundles and the structure of transitive Lie bialgebroids.
Lie Algebroids and Classification Problems in Geometry
2007
We show how one can associate to a given class of finite type G-structures a classifying Lie algebroid. The corresponding Lie groupoid gives models for the different geometries that one can find in the class, and encodes also the different types of symmetry groups.
Generalized Lie Algebroids and Connections over Pair of Diffeomorphic Base Manifolds
Journal of Generalized Lie Theory and Applications, 2013
Extending the definition of Lie algebroid from one base manifold to a pair of diffeomorphic base manifolds, we obtain the generalized Lie algebroid. When the diffeomorphisms used are identities, then we obtain the definition of Lie algebroid. We extend the concept of tangent bundle, and the Lie algebroid generalized tangent bundle is obtained. In the particular case of Lie algebroids, a similar Lie algebroid with the prolongation Lie algebroid is obtained. A new point of view over (linear) connections theory of Ehresmann type on a fiber bundle is presented. These connections are characterized by a horizontal distribution of the Lie algebroid generalized tangent bundle. Some basic properties of these generalized connections are investigated. Special attention to the class of linear connections is paid. The recently studied Lie algebroids connections can be recovered as special cases within this more general framework. In particular, all results are similar with the classical results. Formulas of Ricci and Bianchi type and linear connections of Levi-Civita type are presented.
1 Relative Tangent Spaces and Almost Lie Structures
2008
The aim of this paper is to give three new examples of generalised algebroids and groupoid-like structures, defind in the previous paper [7]. The generalised algebroids include the known definitions of Lie algebroid, prealgebroid and Courant algebroid and the new definition of a generalised prealgebroid. The non-trivial examples that are given in the paper extend those considered in the previous paper [7]. 1 Relative Tangent Spaces and Almost Lie Structures Let (θ,D) be an anchored vector bundle (AVB) ( or a relative tangent space, defined in [3]), where θ = (R, q,M) is a vector bundle and is D : θ → τM a vector bundle morphism of θ, called an anchor (an arrow, or a tangent map). Notice that we denoted as τM = (TM, p,M) the tangent bundle of M . If ξ = (E, π,M) is an other vector bundle on the same base M , then consider the fibered product RE = TE ×TM R = {(x, y) ∈ TE × R : π∗(x) = D(y)} of the differential π∗ : TE → TM and of the given anchor D : R → τM . Let ∆ : RE → TE be the ca...