XXXII Workshop on Geometric Methods in Physics | 30.06-6.07.2013 |

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## Kirill Mackenzie## Approaches to the integrability of Lie algebroidsGiven 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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University of Bialystok |

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