Lie Algebroids, Holonomy and Characteristic Classes

By Rui Loja Fernandes. 2000

282 citations recorded by the source corpus.

Abstract We extend the notion of connection in order to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of a covariant connection. It allows us to define holonomy of the orbit foliation of a Lie algebroid and prove a Stability Theorem. We also introduce secondary or exotic characteristic classes, thus providing invariants which generalize the modular class of a Lie algebroid.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3dbe99c9797d3f63c6fee169eb24dda91a90fdd1

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Citations in the corpus

The geometry of a bi-Lagrangian manifold (2004, background): “ + F(X)(f)Y, where f is a function on M, which proves the second one, thus showing that the tangent bundle TM becomes a Lie algebroid with bracket [−,−]F and anchor map F: TM→ TM. Moreover, Fernandes [27] deﬁned the notion of Lie algebroid connection; in the present situation a pseudo-connection who”
Riemannian Geometry of Lie Algebroids (2008, background): “They have many applications in various part of mathematics (see for instance [4, 5, 6, 14]).”
Lie algebroid structures on double vector bundles and representation theory of Lie algebroids (2008, methodology): “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.”
VB-groupoids and representation theory of Lie groupoids (2010, background): “It is known [2, 3, 6, 7, 11] that the modular class is only the first of a sequence of “secondary” characteristic classes associated to Lie algebroid representations up to homotopy.”
Supergroupoids, double structures, and equivariant cohomology (2006, background): “ connection and isomorphic to H•(BG); however, we do not prove this. We also expect that these “algebroid characteristic classes” should be closely related to those of Crainic and Fernandes [16] [17] [24]. We plan to explore these topics in future work. 4Since we will be using inﬁnitesimal models, w”
Geometry of orbit spaces of proper Lie groupoids (2010, background): “singular foliation: ﬁrst of all, by the local splitting theorem [Fer, Thm 1.1], the distribution is smooth. Second, the maximal integral submanifolds L of D can be described in terms of “Apaths” (cf. [Fer]) as follows: we say that x, y ∈ G0 are A-equivalent, written x ∼A y, if there exists an A-path”
Geometric structures as deformed infinitesimal symmetries (2004, background): “Lie algebroid formalism suggests a generalization of aﬃne connections called g -connections , studied systematically in [8] (where they are the A -derivatives associated with A -connections ).”
Grothendieck Groups of Poisson Vector Bundles (2000, background): “Following Fernandes [ Fe2 ], we refer to such a path as an A-path.”
Generalized Complex Geometry (2004, background): “r derivative, which takes U to U 1 = L∗ ⊗U, is an example of a Lie algebroid connection on the line bundle Uwith respect to the Lie algebroid L. The notion of Lie algebroid connection is developed in [17]. Deﬁnition 3.40 (Lie algebroid connection). Let Lbe a Lie algebroid, Ea vector bundle and D: C∞”