|XXXII Workshop on Geometric Methods in Physics||30.06-6.07.2013|
Participants of Workshop
Participants of School
Approaches to the integrability of Lie algebroids
Given a principal bundle P(M,G), the G-invariant vector fields on P are sections of a vector bundle on M which is the central term of a short exact sequence, the Atiyah sequence of P(M,G). This is a transitive Lie algebroid and by extending the classical geometric prequantization technique one can find, for any transitive Lie algebroid, a Cech class which determines whether the Lie algebroid is the Atiyah sequence of a principal bundle. We will describe this construction in the first lecture.
For general Lie algebroids the problem of integrability is very much more difficult. In the second lecture we will outline the approach of Cattaneo-Felder-Crainic-Fernandes which starts from Duistermaat's correspondence between paths in a Lie group and paths in its Lie algebra.
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