scholarly journals Stacky Hamiltonian Actions and Symplectic Reduction

2020 ◽  
Author(s):  
Benjamin Hoffman ◽  
Reyer Sjamaar
Keyword(s):  
Lie Group ◽  
Symplectic Reduction ◽  
Reduction Theorem ◽  
Convexity Theorem ◽  
Hamiltonian Action ◽  
Hamiltonian Actions

Abstract We introduce the notion of a Hamiltonian action of an étale Lie group stack on an étale symplectic stack and establish versions of the Kirwan convexity theorem, the Meyer–Marsden–Weinstein symplectic reduction theorem, and the Duistermaat–Heckman theorem in this context.

Rigidity of Hamiltonian Actions

2003 ◽  
Vol 46 (2) ◽  
pp. 277-290 ◽  
Author(s):  
Frédéric Rochon
Keyword(s):  
General Theory ◽  
Lie Group ◽  
Geometric Analysis ◽  
Smooth Action ◽  
Symplectic Action ◽  
Hamiltonian Actions

AbstractThis paper studies the following question: Given an ω′-symplectic action of a Lie group on a manifoldMwhich coincides, as a smooth action, with a Hamiltonian ω-action, when is this action a Hamiltonian ω′-action? Using a result of Morse-Bott theory presented in Section 2, we show in Section 3 of this paper that such an action is in fact a Hamiltonian ω′-action, provided thatM is compact and that the Lie group is compact and connected. This result was first proved by Lalonde-McDuff-Polterovich in 1999 as a consequence of a more general theory that made use of hard geometric analysis. In this paper, we prove it using classical methods only.

scholarly journals GEOMETRIC QUANTIZATION OF HAMILTONIAN ACTIONS OF LIE ALGEBROIDS AND LIE GROUPOIDS

2007 ◽  
Vol 04 (03) ◽  
pp. 389-436 ◽  
Author(s):  
ROGIER BOS
Keyword(s):  
Geometric Quantization ◽  
Symplectic Manifolds ◽  
Lie Algebroids ◽  
Unitary Representations ◽  
Reduction Theorem ◽  
Orbit Method ◽  
Lie Algebroid ◽  
Lie Groupoids ◽  
Quantization Procedure ◽  
Hamiltonian Actions

We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose, we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden–Weinstein quotient for our setting and prove a "quantization commutes with reduction" theorem. We explain how our geometric quantization procedure relates to a possible orbit method for Lie groupoids. Our theory encompasses the geometric quantization of symplectic manifolds, Hamiltonian Lie algebra actions, actions of bundles of Lie groups, and foliations, as well as some general constructions from differential geometry.

FIBER INTEGRATION OF THE COUPLING CLASS

2013 ◽  
Vol 10 (08) ◽  
pp. 1360016
Author(s):  
ANDRÉS VIÑA
Keyword(s):  
Symplectic Manifold ◽  
Lie Group ◽  
Hamiltonian Action

Given a Hamiltonian action of a Lie group (not necessarily compact) on a symplectic manifold M, by means of the integration over M of the coupling class associated with this action, we define an element which is invariant under deformations of the action.

scholarly journals Integrable geodesic flows on homogeneous spaces

1981 ◽  
Vol 1 (4) ◽  
pp. 495-517 ◽  
Author(s):  
A. Thimm
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
First Integrals ◽  
Geodesic Flows ◽  
Hamiltonian Action ◽  
Grassmannian Manifold ◽  
Tangent Bundles ◽  
Integrable Geodesic Flows

AbstractA method is exposed which allows the construction of families of first integrals in involution for Hamiltonian systems which are invariant under the Hamiltonian action of a Lie group G. This is applied to invariant Hamiltonian systems on the tangent bundles of certain homogeneous spaces M = G/K. It is proved, for example, that every such invariant Hamiltonian system is completely integrable if M is a real or complex Grassmannian manifold or SU(n + 1)/SO(n + 1) or a distance sphere in ℂPn+1. In particular, the geodesic flows of these homogeneous spaces are integrable.

scholarly journals A Hofer-Type Norm of Hamiltonian Maps on Regular Poisson Manifold

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Dawei Sun ◽  
Zhenxing Zhang
Keyword(s):  
Lie Group ◽  
Hamiltonian Path ◽  
Sufficient Conditions ◽  
Poisson Manifold ◽  
Compact Lie Group ◽  
Hamiltonian Action ◽  
Hamiltonian Map

We define a Hofer-type norm for the Hamiltonian map on regular Poisson manifold and prove that it is nondegenerate. We show that theL1,∞-norm and theL∞-norm coincide for the Hamiltonian map on closed regular Poisson manifold and give some sufficient conditions for a Hamiltonian path to be a geodesic. The norm between the Hamiltonian map and the induced Hamiltonian map on the quotient of Poisson manifold(M,{·,·})by a compact Lie group Hamiltonian action is also compared.

scholarly journals On Gibbs States of Mechanical Systems with Symmetries

2020 ◽  
Vol 57 ◽  
pp. 45-85
Author(s):  
Charles-Michel Marle ◽  
Keyword(s):  
Half Plane ◽  
Symplectic Manifold ◽  
Lie Group ◽  
Mechanical Systems ◽  
Companion Paper ◽  
Symplectic Manifolds ◽  
Gibbs States ◽  
Hamiltonian Action ◽  
French Mathematician ◽  
Group Of Symmetries

The French mathematician and physicist Jean-Marie Souriau studied Gibbs states for the Hamiltonian action of a Lie group on a symplectic manifold and considered their possible applications in Physics and Cosmology. These Gibbs states are presented here with detailed proofs of all the stated results. A companion paper to appear will present examples of Gibbs states on various symplectic manifolds on which a Lie group of symmetries acts by a Hamiltonian action, including the Poincar\'e disk and the Poincaré half-plane.

scholarly journals Hamiltonian actions of unipotent groups on compact K\"ahler manifolds

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Daniel Greb ◽  
Christian Miebach
Keyword(s):  
Lie Groups ◽  
Invariant Theory ◽  
Symplectic Reduction ◽  
Stability Conditions ◽  
Unipotent Group ◽  
Geometric Invariant ◽  
Projective Varieties ◽  
Final Version ◽  
Unipotent Groups ◽  
Hamiltonian Actions

We study meromorphic actions of unipotent complex Lie groups on compact K\"ahler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric quotients that carry compactifiable K\"ahler structures obtained by symplectic reduction. The relation of our complex-analytic theory to the work of Doran--Kirwan regarding the Geometric Invariant Theory of unipotent group actions on projective varieties is discussed in detail. Comment: v2: 30 pages, final version as accepted by EPIGA

scholarly journals FORMAL FROBENIUS MANIFOLD STRUCTURE ON EQUIVARIANT COHOMOLOGY

1999 ◽  
Vol 01 (04) ◽  
pp. 535-552 ◽  
Author(s):  
HUAI-DONG CAO ◽  
JIAN ZHOU
Keyword(s):  
Lie Group ◽  
Equivariant Cohomology ◽  
Kähler Manifold ◽  
Frobenius Manifold ◽  
Algebra Structure ◽  
Compact Lie Group ◽  
Hamiltonian Action ◽  
Manifold Structure ◽  
Kahler Manifold ◽  
Cartan Model

For a closed Kähler manifold with a Hamiltonian action of a connected compact Lie group by holomorphic isometries, we construct a formal Frobenius manifold structure on the equivariant cohomology by exploiting a natural DGBV algebra structure on the Cartan model.

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