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BIRS Workshop Lecture Videos

Lie groupoids which give rise to G-structures Struchiner, Ivan


The infinitesimal data attached to a (“finite type” class of) G-structures with connections are its structure equations. Such structure equations give rise to Lie algebroids endowed with extra geometric information following from the fact that they come from G-structures. For example, the Lie algebroid is trivial as a vector bundle, its bracket encodes the Lie bracket and the natural representation on $R^n$ of the Lie algebra of the structure group G of the G-structure, the Lie algebroid comes equipped an action of G by inner Lie algebroid automorphisms, etc… Conversely, given a Lie algebroid as above (called a G-algebroid), a natural question is that of finding G-structures which correspond via differentiation to the Lie algebroid. This integration problem is known as “Cartan’s Realization Problem for G-Structures”. In this talk I will show that if a G-algebroid is integrable by a Lie groupoid endowed with an action of G, then each s-fiber of the groupoid can be identified with the total space of a G-structure with connection solving the realization problem. If time permits I will also explain the obstructions for finding such “G-integrations” of G-algebroids. The talk will be based on joint work with prof. Rui Loja Fernandes

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