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

Fonte: Centro de Matemática da Universidade de Coimbra
Publicador: Centro de Matemática da Universidade de Coimbra

Tipo: Pré-impressão

Português

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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

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## Reduction of Lie algebroid structures

Fonte: Centro de Matemática da Universidade de Coimbra
Publicador: Centro de Matemática da Universidade de Coimbra

Tipo: Pré-impressão

Português

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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.

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

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Tese de Doutorado
Formato: application/pdf

Publicado em 23/01/2009
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#Algebroide de Lie#Lie#Simetrias de#Geometria diferencial#Lie algebroid#Lie symmetries#Differential geometry

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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematical Physics#High Energy Physics - Theory#Mathematics - Algebraic Geometry#Mathematics - Differential Geometry

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

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## Differential calculus on a Lie algebroid and Poisson manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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## Modular classes of Lie algebroid morphisms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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## Vertex Algebras and the Equivariant Lie Algebroid Cohomology

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/05/2013
Português

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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.

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## Characteristic Classes of Lie Algebroid Morphisms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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## Killing sections and sigma models with Lie algebroid targets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/06/2015
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#Mathematics - Differential Geometry#High Energy Physics - Theory#Mathematical Physics#53C15, 53C07, 53D17, 53C22, 53Z05

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

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## Dirac actions and Lu's Lie algebroid

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/12/2014
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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

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## Lie algebroid modules and representations up to homotopy

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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## Lie Algebroid Yang Mills with Matter Fields

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/08/2009
Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/08/2012
Português

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Topology#Mathematics - Category Theory#55R40, 57R20

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

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## Gravity from Lie algebroid morphisms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/10/2003
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#High Energy Physics - Theory#General Relativity and Quantum Cosmology#Mathematics - Differential Geometry

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

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## Nonabelian holomorphic Lie algebroid extensions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Symplectic Geometry#Mathematics - Algebraic Geometry#4F05, 14F40, 32L10, 55N25, 55N91, 55R20

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

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## Poisson and symplectic functions in Lie algebroid theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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.

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## Lie algebroid structures on a class of affine bundles

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/01/2002
Português

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We introduce the notion of a Lie algebroid structure on an affine bundle
whose base manifold is fibred over the real numbers. It is argued that this is
the framework which one needs for coming to a time-dependent generalization of
the theory of Lagrangian systems on Lie algebroids. An extensive discussion is
given of a way one can think of forms acting on sections of the affine bundle.
It is further shown that the affine Lie algebroid structure gives rise to a
coboundary operator on such forms. The concept of admissible curves and
dynamical systems whose integral curves are admissible, brings an associated
affine bundle into the picture, on which one can define in a natural way a
prolongation of the original affine Lie algebroid structure.; Comment: 28 pages

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