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Convexity and Loop Groups

  • Michael Francis Atiyah
  • Andrew Nicholas Pressley
Part of the Progress in Mathematics book series (PM, volume 36)

Abstract

The purpose of this paper is to extend certain convexity results associated with compact Lie groups to an infinite-dimensional setting, in which the Lie group is replaced by the corresponding loop group. To recall the finite-dimensional results which we shall generalize let G be a simply connected, compact Lie group, T a maximal torus of G and W its Weyl group. Consider the adjoint action of G on its Lie algebra L(G) and fix a G-invariant metric on L(C) so that we can define orthogonal projection. A result of Kostant [8] describes the images of the G-orbits in L(G) under the orthogonal projection onto L(T). To state it, recall that such G-orbits correspond to W-orbits in L(T). Then Kostant's result is:

(1.1). The orthogonal projection of a G-orbit onto L(T) coincides with the convex hull of the corresponding W-orbit.

Keywords

Convex Hull Weyl Group Symplectic Form Maximal Torus Loop Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1983

Authors and Affiliations

  • Michael Francis Atiyah
    • 1
  • Andrew Nicholas Pressley
    • 1
  1. 1.Mathematical InstituteOxfordEngland

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