Symplectic Group Actions and Covering Spaces (original) (raw)
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The reduced spaces of a symplectic Lie group action
Annals of Global Analysis and Geometry, 2006
There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino . In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.
An abstract setting for hamiltonian actions
Monatshefte für Mathematik, 2010
In this paper we develop an abstract setup for hamiltonian group actions as follows: Starting with a continuous 2-cochain ω on a Lie algebra h with values in an h-module V , we associate subalgebras sp(h, ω) ⊇ ham(h, ω) of symplectic, resp., hamiltonian elements. Then ham(h, ω) has a natural central extension which in turn is contained in a larger abelian extension of sp(h, ω). In this setting, we study linear actions of a Lie group G on V which are compatible with a homomorphism g → ham(h, ω), i.e., abstract hamiltonian actions, corresponding central and abelian extensions of G and momentum maps J : g → V .
The stratified spaces of a symplectic lie group action
Reports on Mathematical Physics, 2006
There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino . In this case it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.
Poisson and Symplectic Structures, Hamiltonian Action, Momentum and Reduction
arXiv: Differential Geometry, 2020
This manuscript is essentially a collection of lecture notes which were given by the first author at the Summer School Wisla-2019, Poland and written down by the second author. As the title suggests, the material covered here includes the Poisson and symplectic structures (Poisson manifolds, Poisson bi-vectors and Poisson brackets), group actions and orbits (infinitesimal action, stabilizers and adjoint representations), moment maps, Poisson and Hamiltonian actions. Finally, the phase space reduction is also discussed. The very last section introduces the Poisson-Lie structures along with some related notions. This text represents a brief review of a well-known material citing standard references for more details. The exposition is concise but pedagogical. The Authors believe that it will be useful as an introductory exposition for students interested in this specific topic.
SpringerBriefs in Mathematics
In this chapter, we will define Hamiltonian flows, Hamiltonian actions and moment maps. The layout of the chapter is as follows. In Sect. 2.1 we recall the original example of a Hamiltonian flow, namely, Hamilton's equations. In Sect. 2.2, we will start by understanding what Hamiltonian vector fields and Hamiltonian functions are. In Sect. 2.3, we will introduce a bracket on the set of smooth functions on a symplectic manifold which will satisfy the Jacobi identity and will make the former into a Lie algebra. We will see some examples of such vector fields on S 2 and the 2-torus. In the final section (Sect. 2.4), we will define a moment map and will list some conditions which will guarantee the existence of moment maps and other conditions which guarantee their uniqueness.
On certain symplectic circle actions
Journal of Symplectic Geometry, 2005
In this work we use localization formulas in equivariant cohomology to show that some symplectic actions on 6-dimensional manifolds with a finite fixed point set must be Hamiltonian. Moreover, we show that their fixed point data (number of fixed points and their isotropy weights) is the same as in S 2 × S 2 × S 2 equipped with a diagonal circle action, and we compute their cohomology rings.
Structure of symplectic Lie groups and momentum map
Tohoku Mathematical Journal, 2015
We describe the structure of the Lie groups endowed with a leftinvariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber.This description is particularly nice if the group is Hamiltonian, that is, if the left canonical action of the group on itself is Hamiltonian. The principal tool used for our description is a canonical affine structure associated with the symplectic form. We also characterize the Hamiltonian symplectic Lie groups among the connected symplectic Lie groups. We specialize our principal results to the cases of simply connected Hamiltonian symplectic nilpotent Lie groups or Frobenius symplectic Lie groups. Finally we pursue the study of the classical affine Lie group as a symplectic Lie group.
A Universal Reduction Procedure for Hamiltonian Group Actions
Mathematical Sciences Research Institute Publications, 1991
We give a universal method of inducing a Poisson structure on a singular reduced space from the Poisson structure on the orbit space for the group action. For proper actions we show that this reduced Poisson structure is nondegenerate. Furthermore, in cases where the Marsden-Weinstein reduction is well-defined, the action is proper, and the preimage of a coadjoint orbit under the momentum mapping is closed, we show that universal reduction and Marsden-Weinstein reduction coincide. As an example, we explicitly construct the reduced spaces and their Poisson algebras for the spherical pendulum.
The symplectic reduced spaces of a Poisson action
Comptes Rendus Mathematique, 2002
During the last thirty years, symplectic or Marsden-Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the presence of a momentum map. In this note we show that the symplectic reduction phenomenon has much deeper roots. More specifically, we will find symplectically reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. On other words, the right category to obtain symplectically reduced spaces is that of Poisson manifolds acted canonically upon by a Lie group.