Equivariant Momentum Map Research Articles

Properties of gradient maps associated with action of real reductive group

Let [Formula: see text] be a Kähler manifold and let [Formula: see text] be a compact connected Lie group with Lie algebra [Formula: see text] acting on [Formula: see text] and preserving [Formula: see text]. We assume that the [Formula: see text]-action extends holomorphically to an action of the complexified group [Formula: see text] and the [Formula: see text]-action on [Formula: see text] is Hamiltonian. Then there exists a [Formula: see text]-equivariant momentum map [Formula: see text]. If [Formula: see text] is a closed subgroup such that the Cartan decomposition [Formula: see text] induces a Cartan decomposition [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] is the Lie algebra of [Formula: see text], there is a corresponding gradient map [Formula: see text]. If [Formula: see text] is a [Formula: see text]-invariant compact and connected real submanifold of [Formula: see text] we may consider [Formula: see text] as a mapping [Formula: see text] Given an [Formula: see text]-invariant scalar product on [Formula: see text], we obtain a Morse like function [Formula: see text] on [Formula: see text]. We point out that, without the assumption that [Formula: see text] is a real analytic manifold, the Lojasiewicz gradient inequality holds for [Formula: see text]. Therefore, the limit of the negative gradient flow of [Formula: see text] exists and it is unique. Moreover, we prove that any [Formula: see text]-orbit collapses to a single [Formula: see text]-orbit and two critical points of [Formula: see text] which are in the same [Formula: see text]-orbit belong to the same [Formula: see text]-orbit. We also investigate convexity properties of the gradient map [Formula: see text] in the Abelian case. In particular, we study two-orbit variety [Formula: see text] and we investigate topological and cohomological properties of [Formula: see text].

Point vortices on the hyperbolic plane

We investigate the dynamical system of point vortices on the hyperboloid. This system has non-compact symmetry SL(2, R) and a coadjoint equivariant momentum map. The relative equilibrium conditions are found and the trajectories of relative equilibria with non-zero momentum value are described. We also provide the classification of relative equilibria and the stability criteria for a number of cases, focusing on 2 and 3 vortices. Unlike the system on the sphere, this system has relative equilibria with non-compact momentum isotropy subgroup, and these are used to illustrate the different stability types of relative equilibria.

The degenerate C. Neumann system I: symmetry reduction and convexity

Abstract The C. Neumann system describes a particle on the sphere S n under the influence of a potential that is a quadratic form. We study the case that the quadratic form has ℓ +1 distinct eigenvalues with multiplicity. Each group of m σ equal eigenvalues gives rise to an O(m σ)-symmetry in configuration space. The combined symmetry group G is a direct product of ℓ + 1 such factors, and its cotangent lift has an Ad*-equivariant momentum mapping. Regular reduction leads to the Rosochatius system on S ℓ, which has the same form as the Neumann system albeit for an additional effective potential. To understand how the reduced systems fit together we use singular reduction to construct an embedding of the reduced Poisson space T*S n/G into ℝ3ℓ+3. The global geometry is described, in particular the bundle structure that appears as a result of the superintegrability of the system. We show how the reduced Neumann system separates in elliptical-spherical co-ordinates. We derive the action variables and frequencies as complete hyperelliptic integrals of genus ℓ. Finally we prove a convexity result for the image of the Casimir mapping restricted to the energy surface.

Geometric quantization, reduction and decomposition of group representations

We consider a Hamiltonian action of a connected group G on a symplectic manifold (P, ω) with an equivariant momentum map \(J : P \rightarrow {\mathfrak{g}^{*}}\) and its quantization in terms of a Kahler polarization which gives rise to a unitary representation \({\mathcal{U}}\) of G on a Hilbert space \({\mathcal{H}}\). If O is a co-adjoint orbit of G quantizable with respect to a Kahler polarization, we describe geometric quantization of algebraic reduction of J−1(O). We show that the space of normalizable states of quantization of algebraic reduction of J−1(O) gives rise to a projection operator onto a closed subspace of \({\mathcal{H}}\) on which \({\mathcal{U}}\) is unitarily equivalent to a multiple of the irreducible unitary representation of G corresponding to O. This is a generalization of the results of Guillemin and Sternberg obtained under the assumptions that G and P are compact and that the action of G on P is free. None of these assumptions are needed here.

Phase Space Reduction of Star Products on Cotangent Bundles

In this paper we construct star products on Marsden-Weinstein reduced spaces in case both the original phase space and the reduced phase space are (symplectomorphic to) cotangent bundles. Under the assumption that the original cotangent bundle $T^*Q$ carries a symplectique structure of form $\omega_{B_0}=\omega_0 + \pi^*B_0$ with $B_0$ a closed two-form on $Q$, is equipped by the cotangent lift of a proper and free Lie group action on $Q$ and by an invariant star product that admits a $G$-equivariant quantum momentum map, we show that the reduced phase space inherits from $T^*Q$ a star product. Moreover, we provide a concrete description of the resulting star product in terms of the initial star product on $T^*Q$ and prove that our reduction scheme is independent of the characteristic class of the initial star product. Unlike other existing reduction schemes we are thus able to reduce not only strongly invariant star products. Furthermore in this article, we establish a relation between the characteristic class of the original star product and the characteristic class of the reduced star product and provide a classification up to $G$-equivalence of those star products on $(T^*Q,\omega_{B_0})$, which are invariant with respect to a lifted Lie group action. Finally, we investigate the question under which circumstances `quantization commutes with reduction' and show that in our examples non-trivial restrictions arise.

Induction of Hamiltonian Poisson actions

We propose a Poisson–Lie analog of the symplectic induction procedure, using an appropriate Poisson generalization of the reduction of symplectic manifolds with symmetry. Having as basic tools the equivariant momentum maps of Poisson actions, the double group of a Poisson–Lie group and the reduction of Poisson manifolds with symmetry, we show how one can induce a Poisson action admitting an equivariant momentum map. We prove that, under certain conditions, the dressing orbits of a Poisson–Lie group can be obtained by Poisson induction from the dressing orbits of a Poisson–Lie subgroup

EQUIVARIANT POISSON COHOMOLOGY AND A SPECTRAL SEQUENCE ASSOCIATED WITH A MOMENT MAP

We introduce and study a new spectral sequence associated with a Poisson group action on a Poisson manifold and an equivariant momentum mapping. This spectral sequence is a Poisson analog of the Leray spectral sequence of a fibration. The spectral sequence converges to the Poisson cohomology of the manifold and has the E2-term equal to the tensor product of the cohomology of the Lie algebra and the equivariant Poisson cohomology of the manifold. The latter is defined as the equivariant cohomology of the multi-vector fields made into a G-differential complex by means of the momentum mapping. An extensive introduction to equivariant cohomology of G-differential complexes is given including some new results and a number of examples and applications are considered.

MOMENTUM MAPPINGS AND POISSON COHOMOLOGY

We analyze the question of existence and uniqueness of equivariant momentum mappings for Poisson actions of Poisson Lie groups. A necessary and sufficient condition for the equivariant momentum mapping to be unique is given. The existence problem is solved under some extra hypotheses, for example, when the action preserves the Poisson structure. In this case, the problem is closely related to the triviality of the induced group action on the Poisson cohomology. This action is shown to be trivial whenever the group is compact or semisimple. Conceptually, these results rely upon a version of “Poisson calculus” developed here to make one-forms on a Poisson manifold induce a “flow” preserving the Poisson structure. In the general case, obstructions to the existence of an infinitesimal version of an equivariant momentum mapping are found. Using Lie algebra cohomology with coefficients in Fréchet modules, we show that the obstructions vanish, and the infinitesimal mapping exists, when the group is compact semisimple. We also prove the rigidity of compact group actions preserving the Poisson structure on a compact manifold and calculate the Poisson cohomology of the Poisson homogeneous space [Formula: see text].

Complexification of gauge theories

For the case of a first-class constrained system with equivariant momentum map, we study the conditions under which the double process of reducing to the constraint surface and dividing out by the group of gauge transformations G is equivalent to the single process of dividing out the initial phase space by the complexification G C of G. For the particular case of a phase space action that is the lift of a configuration space action, conditions are found under which, in finite dimensions, the physical phase space of a gauge system with first-class constraints is diffeomorphic to a manifold imbedded in the physical configuration space of the complexified gauge system. Similar conditions are shown to hold for the infinite-dimensional example of Yang-Mills theories. As a physical application we discuss the adequateness of using holomorphic Wilson loop variables as (generalized) global coordinates on the physical phase space of Yang-Mills theory.

Hamiltonian group representations and phase actions

An action of a group G on a set S is a (surjective) composition law ƒ: G × S → S, associative with respect to the group multiplication m: G × G → G. We show that an action of a Lie group G on a symplectic manifold P together with an equivariant momentum mapping can be characterized as a symplectic reduction ϱ: T* G × P → P, associative with respect to Pm, where P is the phase functor [1]. Applications to a study of symplectic analogues of pseudogroups [4] (called also quantum groups [5]) and their representations will be presented in subsequent publications.

A note on the covariant anomaly as an equivariant momentum mapping

We show that there is a natural gauge invariant presymplectic structureω on the spaceA of all vector potentials. The covariant axial anomaly\(\tilde G\) is found to be the essentially unique infinitestimally equivariant momentum mapping for the action of the group of gauge transformations on (A,ω). The infinitesimal equivariance of\(\tilde G\) is shown to be equivalent to the Wess-Zumino consistency condition for the consistent axial anomalyG. We also show that theX operator of Bardeen and Zumino, which relatesG and\(\tilde G\), corresponds to the one-form (onA) of the presymplectic structureω.

The Existence of a Coadjoint Equivariant Momentum Mapping for a Semidirect Product

The Existence of a Coadjoint Equivariant Momentum Mapping for a Semidirect Product

The existence of a coadjoint equivariant momentum mapping for a semidirect product

We consider a symplectic action of a group G G on a symplectic manifold P P , which admits a momentum mapping. Assume that G G is a semidirect product of G 1 {G_1} by G 2 {G_2} . We prove that if the symplectic action of G 1 {G_1} has a coadjoint equivariant momentum mapping, and if H 1 ( u n k 1 : R ) = H 2 ( u n k 2 : R ) = 0 {H^1}(un{k_1}:{\mathbf {R}}) = {H^2}(un{k_2}:{\mathbf {R}}) = 0 , then the symplectic action of G G has a coadjoint equivariant momentum mapping, where u n k 1 un{k_1} and u n k 2 un{k_2} are the Lie algebras of G 1 {G_1} and G 2 {G_2} respectively.