22 research outputs found
Crossed modules and the integrability of Lie brackets
We show that the integrability obstruction of a transitive Lie algebroid
coincides with the lifting obstruction of a crossed module of groupoids
associated naturally with the given algebroid. Then we extend this result to
general extensions of integrable transitive Lie algebroids by Lie algebra
bundles. Such a lifting obstruction is directly related with the classification
of extensions of transitive Lie groupoids. We also give a classification of
such extensions which differentiates to the classification of transitive Lie
algebroids discussed in \cite{KCHM:new}.Comment: 34 pages, revised version. New abstract and introduction. Added
examples and remark
On a remark by Alan Weinstein
Alan Weinstein remarked that, working in the framework of diffeology, a
construction from Noncommutative Differential Geometry might provide the
non-trivial representations required for the geometric quantisation of a
symplectic structure which is not integral. In this note we show that the
construction we gave with P. Antonini does indeed provide non-trivial
representations.Comment: 16 pages, to appear in Contemporary Mathematics, Proceedings of the
AMS-EMS-SMF conference, special section on Diffeology, Grenoble 202
Integrable lifts for transitive Lie algebroids
Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid is the quotient of a finite-dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an \u201cAlmeida\u2013Molino\u201d integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a \u201cde Rham\u201d integrable lift for any given transitive Abelian Lie algebroid
Integration of Singular Subalgebroids
We establish a Lie theory for singular subalgebroids, objects which
generalize singular foliations to the setting of Lie algebroids. First we carry
out the longitudinal version of the theory. For the global one, a guiding
example is provided by the holonomy groupoid, which carries a natural
diffeological structure in the sense of Souriau. We single out a class of
diffeological groupoids satisfying specific properties and introduce a
differentiation-integration process under which they correspond to singular
subalgebroids. In the regular case, we compare our procedure to the usual
integration by Lie groupoids. We also specify the diffeological properties
which distinguish the holonomy groupoid from the graph.Comment: 65 pages, 1 figur