Year: 2006
Type: article
Abstract: For a compact, connected, simply-connected Lie group G , the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H 1 -Sobolev maps S 1 !G: The geometry of LG and its homogeneous spaces is related to representation theory and has been extensively studied.The space of based loops .G/ is an example of a homogeneous space of LG and has a natural Hamiltonian T S 1 action, where T is the maximal torus of G .We study the moment map for this action, and in particular prove that its regular level sets are connected.This result is as an infinite-dimensional analogue of a theorem of Atiyah that states that the preimage of a moment map for a Hamiltonian torus action on a compact symplectic manifold is connected.In the finite-dimensional case, this connectivity result is used to prove that the image of the moment map for a compact Hamiltonian T -space is convex.Thus our theorem can also be viewed as a companion result to a theorem of Atiyah and Pressley, which states that the image ..G// is convex.We also show that for the energy functional E , which is the moment map for the S 1 rotation action, each non-empty preimage is connected.
Source: Geometry & Topology
Authors: Megumi Harada, Tara S Holm, Lisa C Jeffrey, Augustin-Liviu Mare
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W1973872457