The moment map revisited
type: Item
instance of: scholarly article
title: The moment map revisited
zbMATH Open document ID: 1091.53054
author: Shlomo Sternberg
author: Victor Guillemin
published in: Journal of Differential Geometry
publication date: 2006-02-07T00:00:00Z
review text: A Hamiltonian action of a Lie group on a symplectic manifold gives rise to a symplectomorphism group, and one can think of a corresponding moment map as being associated with that symplectomorphism group. In the paper under review a general notion of moment map is described, which allows one to speak about moment maps associated with families (not necessarily groups) of symplectomorphisms. The general setting is provided by a symplectic manifold \((M,\omega)\), a fibration \(\pi\colon Z\to S\), and a smooth map \(G\colon Z\to M\). For any \(s\in S\) one denotes \(Z_s=\pi^{-1}(s)\), \(g_s=G| _{Z_s}: Z_s\to M\), \(\Lambda_s=g_s(Z_s)\), and one assumes that the mapping \(g_s\) is a Lagrangian embedding. Then a moment map is a lifting \(\Phi: Z\to T^*S\) of \(\pi: Z\to S\) such that for all \(s\in S\) and \(z\in Z_s\) the mapping \(\Phi_s=\Phi| _{Z_s}: Z_s\to T_s^*S\) satisfies the following condition: The dual (i.e., transpose) of the differential \((d\Phi_s)_z: T_z(Z_s)\to T_s^*S\) coincides with the linear map \(T_sS\to T_z^*(Z_s)\) induced by \(dG_z: T_zZ\to T_{G(z)}M\) by factoring to a linear map \(T_zZ/T_z(Z_s)\to T_{G(z)}M/T_{G(z)}(\Lambda_s)\) and then using the isomorphisms \(T_zZ/T_z(Z_s)\simeq T_sS\) (defined by the fibration \(\pi\)) and \(T_{G(z)}M/T_{G(z)}(\Lambda_s)\!\simeq T_{G(z)}^*(\Lambda_s)\simeq T_z^*(Z_s)\) (defined by the Lagrangian embedding \(g_s=G| _{Z_s}: Z_s\to\Lambda_s\hookrightarrow M\) along with the symplectic form \(\omega\) of \(M\)). This formalism can be applied in particular to families (indexed by \(S\)) of symplectomorphisms of \(M\), by working with \(S\times M\) in the role of \(Z\) and with \(M\times M^{-}\) in the role of \(M\) (where \(M^{-}\) stands for \(M\) equipped with the symplectic form \(-\omega\)). The main results of the paper include some basic properties of this generalized moment map. Thus the pull-back by \((G,\Phi): Z\to M\times T^*S\) of the symplectic form of \(M\times T^*\) is the pull-back by \(\pi\) of a c…
reviewed by: Daniel Beltiţă
Mathematics Subject Classification ID: 53D20
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 53D05
zbMATH DE Number: 5004319
zbMATH Keywords: moment map
zbMATH Keywords: symplectomorphism
zbMATH Keywords: Lagrangian embedding
Wikidata QID: Q115198571
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1121540342
OpenAlex ID: W1556959844
DOI: 10.4310/JDG/1121540342
description: scientific article; zbMATH DE number 5004319