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Journal of Lie Theory 26 (2016), No. 2, 535--565
Copyright Heldermann Verlag 2016

Pseudogroups via Pseudoactions: Unifying Local, Global, and Infinitesimal Symmetry

Anthony D. Blaom
10 Huruhi Road, Waiheke Island 1081, Auckland, New Zealand

A multiplicatively closed, horizontal foliation on a Lie groupoid may be viewed as a "pseudoaction" on the base manifold M. A pseudoaction generates a pseudogroup of transformations of M in the same way an ordinary Lie group action generates a transformation group. Infinitesimalizing a pseudoaction, one obtains the action of a Lie algebra on M, possibly twisted. A global converse to Lie's third theorem proven here states that every twisted Lie algebra action is integrated by a pseudoaction. When the twisted Lie algebra action is complete it integrates to a twisted Lie group action, according to a generalization of Palais' global integrability theorem.

Keywords: Lie algebroid, pseudogroup, Cartan connection, Lie algebra, pseudoaction.

MSC: 58H05; 58D19

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