Adjoint and coadjoint orbits of the Euclidean group

Arathoon, Philip and Montaldi, James (2015) Adjoint and coadjoint orbits of the Euclidean group. [MIMS Preprint]

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Abstract

We give a geometric description of the adjoint and coadjoint orbits of the special Euclidean group. We implement the method of little subgroups as introduced by Rawnsley in 1975 and the method of types by Burgoyne and Cushman in 1977 to classify these orbits completely. The orbits are diffeomorphic to affine flag manifolds, whose definition and geometry we also explore. Since coadjoint orbits are naturally symplectic, such manifolds provide us with interesting examples of symplectic homogeneous spaces. As discovered by Cushman and van der Kallen in 2006, we identify a bijection between the coadjoint and adjoint orbits of the Euclidean group. Furthermore, we show that orbits corresponding under this bijection are homotopy equivalent. Whether the bijection for other groups, and especially the Poincare group, preserves homotopy type of orbits remains an open question.

Item Type: MIMS Preprint
Uncontrolled Keywords: symplectic geometry, Lie groups
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 22 Topological groups, Lie groups
Depositing User: Dr James Montaldi
Date Deposited: 03 May 2015
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2292

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