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## Quasi-coordinates from the point of view of Lie algebroid structures

Cariñena, J. F.; Costa, J. M. Nunes da; Santos, Patrícia
Tipo: Pré-impressão
Português
Relevância na Pesquisa
56.62%
In this paper a geometrical description of the Lagrangian dynamics in quasi-coordinates on the tangent bundle, using the Lie algebroid framework, is given. Linear non-holonomic systems on Lie algebroids are solved in local coordinates adapted to the constraints, through generalized methods of the Lagrangian multipliers and of Gibbs-Appell.; PRODEP/5.3/2003; POCI/MAT/ 58452/2004; CMUC/FCT; project BFM-2003-02532

## Reduction of Lie algebroid structures

Cariñena, J. F.; Costa, J. M. Nunes da; Santos, Patrícia
Tipo: Pré-impressão
Português
Relevância na Pesquisa
56.83%
Based on the ideas of Marsden-Ratiu, a reduction method for Lie algebroids is developed in such a way that the canonical projection onto the reduced Lie algebroid is a homomorphism of Lie algebroids. A relation between Poisson reduction and Lie algebroid reduction is established. Reduction of Lie algebroids with symmetry is also studied using this method.; PRODEP/5.3/2003; POCTI/MAT/ 58452/2004; CMUC/FCT, project BFM-2003-02532.

## O algebroide classificante de uma estrutura geometrica; The classifying Lie algebroid of a geometric structure

Ivan Struchiner
Fonte: Biblioteca Digital da Unicamp Publicador: Biblioteca Digital da Unicamp
Tipo: Tese de Doutorado Formato: application/pdf
Relevância na Pesquisa
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O objetivo desta tese é mostrar como utilizar algebróides de Lie e grupóides de Lie para compreender aspectos das teorias de invariantes, simetrias e espaços de moduli de estruturas geométricas de tipo finito. De uma forma geral, podemos descrever tais estruturas como sendo objetos, definidos em uma variedade, que podem ser caracterizados por correferenciais (possivelmente em outra variedade). Exemplos incluem G-estruturas de tipo finito e geometrias de Cartan. Para uma classe de estruturas geométricas de tipo finito cujo espaço de moduli (dos germes) de seus elementos tem dimensão finita, construímos um algebróide de Lie A X, chamado de algebróide de Lie classificante, que satisfaz as seguintes propriedades: 1. Para cada ponto na base X corresponde um germe de uma estrutura geométrica pertencente à classe. 2. Dois destes germes são isomorfos se e somente se eles correspondem ao mesmo ponto de X. 3. A álgebra de Lie de isotropia de A num ponto x é a álgebra de Lie das simetrias infinitesimais da estrutura geométrica correspondente. 4. Se dois germes de estruturas geométricas pertencem à mesma estrutura geométrica global numa variedade conexa, então eles correspondem a pontos na mesma órbita de A em X. Além do mais...

## Metrizability of the Lie algebroid generalized tangent bundle and (generalized) Lagrange $(\rho,\eta)$-spaces

Arcuş, Constantin M.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.9%
A class of metrizable vector bundles in the general framework of generalized Lie algebroids have been presented in the eight reference. Using a generalized Lie algebroid we obtain the Lie algebroid generalized tangent bundle of a vector bundle. This Lie algebroid is a new example of metrizable vector bundle. A new class of Lagrange spaces, called by use, generalized Lagrange (\rho?;\eta?)-space, Lagrange (\rho?;\eta?)-space and Finsler (\rho?;\eta?)-space are presented. In the particular case of Lie algebroids, new and important results are presented. In particular, if all morphisms are identities morphisms, then the classical results are obtained.; Comment: 33 pages

## Boundary coupling of Lie algebroid Poisson sigma models and representations up to homotopy

Velez, Alexander Quintero
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.71%
A general form for the boundary coupling of a Lie algebroid Poisson sigma model is proposed. The approach involves using the Batalin-Vilkovisky formalism in the AKSZ geometrical version, to write a BRST-invariant coupling for a representation up to homotopy of the target Lie algebroid or its subalgebroids. These considerations lead to a conjectural description of topological D-branes on generalized complex manifolds, which includes A-branes and B-branes as special cases.; Comment: 24 pages, no figures; v2: published version

## Differential calculus on a Lie algebroid and Poisson manifolds

Marle, Charles-Michel
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.76%
A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present in this paper the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds.; Comment: 46 pages

## Formality theorems for Hochschild chains in the Lie algebroid setting

Calaque, Damien; Dolgushev, Vasiliy; Halbout, Gilles
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.84%
In this paper we prove Lie algebroid versions of Tsygan's formality conjecture for Hochschild chains both in the smooth and holomorphic settings. In the holomorphic setting our result implies a version of Tsygan's formality conjecture for Hochschild chains of the structure sheaf of any complex manifold and in the smooth setting this result allows us to describe quantum traces for an arbitrary Poisson Lie algebroid. The proofs are based on the use of Kontsevich's quasi-isomorphism for Hochschild cochains of R[[y_1,...,y_d]], Shoikhet's quasi-isomorphism for Hochschild chains of R[[y_1,...,y_d]], and Fedosov's resolutions of the natural analogues of Hochschild (co)chain complexes associated with a Lie algebroid.; Comment: 40 pages, no figures

## Modular classes of Lie algebroid morphisms

Kosmann-Schwarzbach, Yvette; Laurent-Gengoux, Camille; Weinstein, Alan
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.88%
We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, including Morita equivalences, that act on the 1-cohomology, and observe that the relative modular class is a coboundary on the category of Lie algebroids and generalized morphisms with values in the 1-cohomology.; Comment: 33 pages. Dedicated to Bertram Kostant for his eightieth birthday. Minor changes in version 2: Proposition 3.11 added, typos corected

## Vertex Algebras and the Equivariant Lie Algebroid Cohomology

Okumura, Masanari
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
47.06%
A vertex-algebraic analogue of the Lie algebroid complex is constructed, which generalizes the "small" chiral de Rham complex on smooth manifolds. The notion of VSA-inductive sheaves is also introduced. This notion generalizes that of sheaves of vertex superalgebras. The complex mentioned above is constructed as a VSA-inductive sheaf. With this complex, the equivariant Lie algebroid cohomology is generalized to a vertex-algebraic analogue, which we call the chiral equivariant Lie algebroid cohomology. In fact, the notion of the equivariant Lie algebroid cohomology contains that of the equivariant Poisson cohomology. Thus the chiral equivariant Lie algebroid cohomology is also a vertex-algebraic generalization of the equivariant Poisson cohomology. A special kind of complex is introduced and its properties are studied in detail. With these properties, some isomorphisms of cohomologies are developed, which enables us to compute the chiral equivariant Lie algebroid cohomology in some cases. Poisson-Lie groups are considered as such a special case.

## Characteristic Classes of Lie Algebroid Morphisms

Vaisman, Izu
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.76%
We extend R. Fernandes' construction of secondary characteristic classes of a Lie algebroid to the case of a base-preserving morphism between two Lie algebroids. Like in the case of a Lie algebroid, the simplest characteristic class of our construction coincides with the modular class of the morphism.; Comment: Latex, 18 pages, small corrections and a new proposition added

## Killing sections and sigma models with Lie algebroid targets

Bruce, Andrew James
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.76%
We define and examine the notion of a Killing section of a Riemannian Lie algebroid as a natural generalisation of a Killing vector field. We show that the various expression for a vector field to be Killing naturally generalise to the setting of Lie algebroids. As an application we examine the internal symmetries of a class of sigma models for which the target space is a Riemannian Lie algebroid. Critical points of these sigma models are interpreted as generalised harmonic maps.; Comment: 11 pages. Comments welcomed

## Dirac actions and Lu's Lie algebroid

Meinrenken, Eckhard
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.79%
Poisson actions of Poisson Lie groups have an interesting and rich geometric structure. We will generalize some of this structure to Dirac actions of Dirac Lie groups. Among other things, we extend a result of Jiang-Hua-Lu, which states that the cotangent Lie algebroid and the action algebroid for a Poisson action form a matched pair. We also give a full classification of Dirac actions for which the base manifold is a homogeneous space $H/K$, obtaining a generalization of Drinfeld's classification for the Poisson Lie group case.; Comment: 39 pages

## Lie algebroid modules and representations up to homotopy

Mehta, Rajan Amit
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.84%
We establish a relationship between two different generalizations of Lie algebroid representations: representation up to homotopy and Vaintrob's Lie algebroid modules. Specifically, we show that there is a noncanonical way to obtain a representation up to homotopy from a given Lie algebroid module, and that any two representations up to homotopy obtained in this way are equivalent in a natural sense. We therefore obtain a one-to-one correspondence, up to equivalence.; Comment: v3: Final version, to appear in Indag. Math

## Lie Algebroid Yang Mills with Matter Fields

Mayer, Christoph; Strobl, Thomas
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.9%
Lie algebroid Yang-Mills theories are a generalization of Yang-Mills gauge theories, replacing the structural Lie algebra by a Lie algebroid E. In this note we relax the conditions on the fiber metric of E for gauge invariance of the action functional. Coupling to scalar fields requires possibly nonlinear representations of Lie algebroids. In all cases, gauge invariance is seen to lead to a condition of covariant constancy on the respective fiber metric in question with respect to an appropriate Lie algebroid connection. The presentation is kept in part explicit so as to be accessible also to a less mathematically oriented audience.; Comment: 24 pages, accepted for publication in J. Geom. Phys

## Comparison of categorical characteristic classes of transitive Lie algebroid with Chern-Weil homomorphism

Mishchenko, Alexander S.; Li, XiaoYu
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.73%
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 (2005) the construction can be managed as a homotopy functor $\mathcal{TLA}_{\rg}$ from category of smooth manifolds to the transitive Lie algebroids. The functor $\mathcal{TLA}_{\rg}$ associates with each smooth manifold $M$ the set $\mathcal{TLA}_{\rg}(M)$ of all transitive algebroids with fixed structural finite dimensional Lie algebra $\rg$. Hence one can construct a classifying space $\cB_{\rg}$ such that the family of all transitive Lie algebroids with fixed Lie algebra $\rg$ over the manifold $M$ has one-to-one correspondence with the family of homotopy classes of continuous maps $[M,\cB_{\rg}]$: $\mathcal{TLA}_{\rg}(M)\approx [M,\cB_{\rg}].$ 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. As a matter of fact we show that the Chern-Weil homomorphism does not cover all characteristic classes from categorical point of view.; Comment: 13 pages

## Gravity from Lie algebroid morphisms

Strobl, Thomas
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.84%
Inspired by the Poisson Sigma Model and its relation to 2d gravity, we consider models governing morphisms from TSigma to any Lie algebroid E, where Sigma is regarded as d-dimensional spacetime manifold. We address the question of minimal conditions to be placed on a bilinear expression in the 1-form fields, S^ij(X) A_i A_j, so as to permit an interpretation as a metric on Sigma. This becomes a simple compatibility condition of the E-tensor S with the chosen Lie algebroid structure on E. For the standard Lie algebroid E=TM the additional structure is identified with a Riemannian foliation of M, in the Poisson case E=T^*M with a sub-Riemannian structure which is Poisson invariant with respect to its annihilator bundle. (For integrable image of S, this means that the induced Riemannian leaves should be invariant with respect to all Hamiltonian vector fields of functions which are locally constant on this foliation). This provides a huge class of new gravity models in d dimensions, embedding known 2d and 3d models as particular examples.; Comment: 17 pages, Comm.Math.Phys., in print

## Nonabelian holomorphic Lie algebroid extensions

Bruzzo, Ugo; Mencattini, Igor; Rubtsov, Vladimir; Tortella, Pietro
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.76%
We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild-Serre spectral sequence for Lie algebras to the holomorphic Lie algebroid setting. As an application, we show that the hypercohomology of the Atiyah algebroid of a line bundle has a natural Hodge structure.; Comment: 22 pages. v2: 26 pages. The material has been reorganized and the exposition improved. Slightly modified title. v3: Typos corrected, exposition streamlined, introduction rewritten

## Poisson and symplectic functions in Lie algebroid theory

Kosmann-Schwarzbach, Yvette
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.79%
Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between non-degenerate Poisson functions and symplectic functions. We determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector.; Comment: Dedicated to Murray Gerstenhaber and Jim Stasheff, 27 pages, to appear in Progress in Math.; editorial changes, one reference added; v4: change of convention for bidegree

## Lie algebroid structures on double vector bundles and representation theory of Lie algebroids

Gracia-Saz, Alfonso; Mehta, Rajan Amit
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.9%
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.

## Lie algebroid structures on a class of affine bundles

Sarlet, W.; Mestdag, T.; Martinez, E.