Project topics and materials
Get Complete Project Material File(s) Now! »
Lie Groupoids and Integrability of Lie Algebroids
A topological groupoid consists of two topological manifolds G and M, together with two surjective maps α : G → M and β : G → M called the source and the target respectively, and the object inclusion map 1 : M → G. A groupoid can be thought as a group in which not every pair (g, g0) can be composed. Indeed, one defines a composition law on the space G ? G := {(g, g0) ∈ G × G, α(g) = β(g 0)} which satisfies some natural properties. (see [25] or [8] for all details). Let us just recall that any element g ∈ G has a two–sided inverse g−1 such that α(g−1) = β(g), β(g−1) = α(g), g−1g = 1α(g) and gg−1 = 1β(g).
The manifold M is called the base manifold, and G is just called the groupoid. It is often denoted with a double arrow G M. For x, y ∈ M, Gx = α−1(x) ⊂ G is called the α–fibre at x, and Gy = β−1(y) ⊂ G is called the β –fibre at y. One denotes Gxy = Gx ∩ Gy. Gxx is a topological group, called the vertex group, or isotropy group. For each x ∈ M, 1x ∈ G is the identity element of Gxx. A Lie groupoid is a topological groupoid for which G and M are smooth manifolds, α, β and 1 are smooth, and the composition law G ? G → G is also smooth. In this case, obviously, the vertex group is a Lie group.
Transitivity
One can define an action of G on the base M. For x ∈ M the action of an element g ∈ Gx is defined by g · x := β(g). For x, y in the same orbit (i.e. there exists g ∈ Gxy, i.e. an element g ∈ G such that α(g) = x and β(g) = y), the α–fibres Gx and Gy are diffeomorphic closed submanifolds of G. The Lie groupoid is said to be transitive if this action has only one orbit. In this case, all α–fibres are diffeomorphic to each other, and all vertex groups as well. In his book [25] K. Mackenzie calls it also a locally trivial groupoid.
The pair groupoid and the anchor map
Let M be a smooth manifold. The pair groupoid is defined as M ×M M, with α((x, y)) = x, β((x, y)) = y, 1x = (x, x) for any x, y ∈ M. Let G M over M, the anchor map is defined as r : G → M × M from the Lie groupoid to the pair groupoid, with r := α × β. It is easy to see that the anchor map of a transitive Lie groupoid is surjective.
The short exact sequence of Lie groupoids
Let GM be a transitive Lie groupoid, and let H := ker(r). By definition, H = {g ∈ G, ∃ x ∈ M, r(g) = 1x = (x, x)} = ∪xx∈M Hx where Hx := {g ∈ G, r(g) = 1x} = {g ∈ G, x ( g ) = ( g ) = x } = Gx . Thus, ker( ) is a bundle α β r of Lie groups of the form Gx , i.e. a Lie groupoid with equal target and source maps. One has the following short exact sequence of Lie groupoids defining a transitive Lie groupoid: 1 ∪x∈M Gxx i G r M × M 1.
Bisections
A bisection s : M → G of α is a map such that: α ◦ s = idM and β ◦ s : M → M is a diffeomorphism. The map r0 : s 7−→β ◦ s is a morphism of groups, between the group of bisections of G M, denoted Bisect(G) and the group of diffeomorphisms of M. There is a short exact sequence at the level of sections as well, the map r0 playing the role of the anchor (and one will call i0 the injection of the corresponding kernel). Let us show that it is surjective and that its kernel is composed of sections of H.
• Surjectivity: take φ ∈ Diff(M). For any x ∈ M, (x, φ(x)) ∈ M × M, so there exists gx ∈ G such that r(gx) = (x, φ(x)), from the surjectivity of r. Let s be a map from M to G defined by s(x) := gx. Then, by definition r◦s(x) = (x, φ(x)), i.e. (since r = α× β), α◦s = idM and β ◦s = φ ∈ Diff(M), i.e. s is a bisection corresponding to the diffeomorphism φ.
• Kernel: if s ∈ Bisect(G), that means β ◦ s(x) = x = α ◦ s(x), i.e. s(x) ∈ Gxx, i.e. s is a map from M to ∪xGxx. Thus the kernel of this map is H. Thus, one has the following short exact sequence in terms of bisections: 1 (∪x∈M Gxx) i0 Bisect(G) r0 Diff(M) 1.
Lie Algebroid of a Lie Groupoid
Given a Lie groupoid G M, one can define its corresponding Lie algebroid as encoding its infinitesimal data much in the same way one can define the Lie algebra of a Lie group. A Lie algebroid is a vector bundle, thus it will be defined fibre by fibre. The fibre Ax over x ∈ M is defined as the tangent space to the α–fibre α−1 (x) (which is a smooth manifold) at 1x. The Lie algebroid is the union of all these fibres. In other words: A := ∪x∈M T1x α−1(x) (1.60).
The anchor ρ of this Lie algebroid is then defined thanks to the restriction of the tangent map T β : T G → T M to the Lie algebroid. Let us remark that if the groupoid is transitive, then its corresponding Lie algebroid is transitive too.
Condition of Integrability
Now, given a transitive Lie algebroid A, one can wonder under which conditions there exists a transitive Lie groupoid G such that A is the Lie algebroid corres-ponding to G by the latter construction. In these cases, the Lie algebroid will be called integrable. Let us notice that any Lie groupoid which integrates a transi-tive Lie algebroid is necessarily transitive. Thus, the question of integrability is equivalent whether the Lie algebroid is transitive or not.
The integrability condition has been studied quite recently by Crainic and Fer-nandes in the general case in their 2011’s lectures ([8]),d and by Mackenzie in the transitive case ([25]). It is quite technical and we just schetch the idea here. The fact is that, given a Lie algebroid A, it is always possible to construct a topological groupoid W(A), called the Weinstein groupoid. Recall that one has the short exact sequences:
Table of contents :
Notations and Conventions
Interdependence of the Chapters
Introduction (Français)
Introduction (English)
1 Geometric and Algebraic Framework
1.1 Differential Geometry of Smooth Manifolds
1.1.1 Definition
1.1.2 Active Diffeomorphisms
1.1.3 Tangent Space
1.1.4 Vector Bundle over a Manifold
1.1.5 Connections on a Vector Bundle
1.1.6 Forms and Differential Complexes
1.2 Transitive Lie Algebroids
1.2.1 Introduction
1.2.2 Definition
1.2.3 Trivialization
1.2.4 Representations
1.2.5 Differential Structures on transitive Lie algebroids
1.2.6 Geometric Structures on transitive Lie algebroids
1.3 Lie Groupoids and Integrability of Lie Algebroids
1.3.1 Lie Groupoids
1.3.2 Lie Algebroid of a Lie Groupoid
1.3.3 Condition of Integrability
1.4 Integrable Transitive Lie Algebroid, Principal Fibre Bundle and Atiyah Sequence
1.4.1 Principal Fibre Bundle from an Integrable Transitive Lie Algebroid
1.4.2 Atiyah Lie Algebroids
1.4.3 Local Basis of Right-Invariant Vector Fields
1.4.4 Erhesman Connections
1.5 Gauge Theories and Reduction of Gauge Symmetry
1.5.1 Mathematical Content of a Gauge Theory
1.5.2 Dressing Field Method
2 Cartan Geometry and Gravitation Theories
2.1 Klein Geometry
2.1.1 An intuitive introduction
2.1.2 General Case
2.1.3 Reductive Klein geometries
2.1.4 Example: Minkowski Space-Time as a Reductive Klein Geometry
2.1.5 Principal fibre bundle’s point of view
2.1.6 Maurer-Cartan form of a Klein geometry
2.2 Cartan Geometry
2.2.1 Cartan Connection
2.2.2 Curvature and Geometric Interpretation
2.2.3 Reductive Cartan Geometry
2.3 Cartan Equivalence of Geometric Structures and Gravitation Theories
2.3.1 Riemannian manifold, tetrad formulation, and Poincaré Cartan geometry
2.3.2 General Relativity in Poincaré Geometry
2.3.3 Wise Approach to MacDowell Mansouri Gravity
2.4 Conclusion
3 Conformal Gauge Theories
3.1 Introduction: Conformal Symmetry in Physics
3.1.1 Gauge Theories
3.1.2 Causal Structure
3.1.3 Conformal Gravity
3.2 Overview of Conformal Symmetry, different equivalent approache
3.2.1 First definitions
3.2.2 Overview of conformal geometry
3.2.3 Objects usually defined on a conformal manifold
3.2.4 1st-order conformal bundles: CO-structures
3.2.5 2nd–order conformal bundles: prolongation of a CO-structure
3.2.6 The Klein pair (G,H) of conformal geometry and its homogeneous space M0
3.2.7 Conformal 2-frame bundle as a reduction of the 2–frame bundle
3.2.8 Cartan geometries equivalent to conformal structures
3.3 Dressing Field Method applied to Conformal Geometry, a Top Down construction of Tractor and Twistor connections and bundles
3.3.1 Real Representation and Tractors
3.3.2 Complex Representation and Twistors
3.4 Weyl Gravity as a Yang–Mills Type Gauge Theory
4 Formulation of Gauge Theories on Transitive Lie Algebroids and Unified Lagrangians
4.1 Transitive Lie algebroids: additional material
4.1.1 Mixed Local Basis: Definitions
4.1.2 Change of Trivialisation
4.1.3 Transformation of Xloc 2 TLA(U, g) under a change of trivialisation
4.1.4 Tensorial Calculus and Riemannian Geometry on Lie Algebroid
4.1.5 Volume Form
4.1.6 Maximal inner form
4.1.7 Integration
4.1.8 Hodge Star Operator
4.2 Riemannian geometry on A and Einstein–Hilbert––Yang–Mills unified Lagrangian
4.2.1 The Mixed Local Basis
4.2.2 Levi–Civita Connection
4.2.3 Riemann Curvature
4.2.4 Ricci curvature
4.2.5 Scalar Curvature
4.2.6 Adjoint Connection and Parallel Metric on L
4.2.7 Einstein–Hilbert––Yang–Mills Unified Lagrangian
4.2.8 Conclusion
4.3 Generalized connection and Yang–Mills–Higgs unified Lagrangian
4.3.1 Generalized connection
4.3.2 Curvature of a Generalized Connection
4.3.3 Gauge Transformation
4.3.4 Yang–Mills–Higgs Unified Lagrangian
4.3.5 Conclusion
5 Cartan Geometry in the Framework of Transitive Lie Algebroids
5.1 Introduction – Cartan geometry: additional material
5.1.1 The G–principal bundle Q related to a Cartan H–principal bundle P
5.1.2 The Ehresman connection related to a Cartan connection
5.1.3 Ehresman connections onQwhich are Cartan connections on P
5.2 The commutative diagram of transitive Lie algebroids related to a Cartan geometry
5.2.1 Two short exact sequences of Transitive Lie algebroid for a Cartan geometry
5.2.2 The diagram
5.3 Cartan connection as an isomorphism of C1(M)–modules
5.3.1 Definition of a Cartan connection in our approach
5.3.2 Comparison with the bundle definition
5.3.3 Condition for an ordinary connection on G(TQ) to reduce to a Cartan connection $Lie on H(TP)
5.4 Crampin & Saunders’ approach of Cartan geometry via Lie algebroids
5.4.1 The generalized space à la Cartan
5.4.2 The Lie groupoids of Fibre Morphisms
5.4.3 The Lie algebroids
5.4.4 Summary of Approaches
5.4.5 Infinitesimal Cartan Connection
5.5 Metric
5.5.1 Metric ˆg on H(TP)
5.5.2 Equivalent triple on the Lie algebroid exact sequence
5.6 Gravitational Theories
5.6.1 Einstein–Hilbert Action
5.7 Conclusion: Generalized Cartan Connection and Gravity Theory
Conclusion
Bibliography
