PROPER GROUPOIDS AND MOMENTUM MAPS: LINEARIZATION, AFFINITY, AND CONVEXITY GROUPOPROPRES ET APPLICATIONS MOMENT: LIN ´ EARISATION, CARACT ` ERE AFFINE, ET CONVEXIT ´ E

By Nguyen Tien Zung. 2004

74 citations recorded by the source corpus.

Abstract We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural integral affine structure, which makes it into an affine orbifold with locally convex polyhedral boundary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e453df5269e486d7b0afae031fe756d3cae16e0c

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Citations in the corpus

The momentum map in Poisson geometry (2007, result): “If this can be solved in a reasonable way, all our results should be generalizable to such proper G-spaces, using the the slice theorem of Zung [37].”
Lie groupoids and their orbispaces (2012, unknown): “To finish, let us mention that the linearization problem is studied here in the spirit of the original works [27] and [28], and the more recent paper [7].”