A family of poisson structures on hermitian symmetric spaces

We investigate the compatibility of symplectic Kirillov-Kostant-Souriau structure and Poisson-Lie structure on coadjoint orbits of semisimple Lie group. We prove that they are compatible for an orbit compact Lie group iff the orbit is hermitian symmetric space. We prove also the compatibility statement for non-compact hermitian symmetric space. As an example we describe a structure of symplectic leaves onCP ⁿ for this family. These leaves may be considered as a perturbation of Schubert cells. Possible applications to infinite-dimensional examples are discussed.     

Reduction of symplectic manifolds with symmetry

We give a unified framework for the construction of symplectic manifolds from systems with symmetries. Several physical and mathematical examples are given; for instance, we obtain Kostant’s result on the symplectic structure of the orbits under the coadjoint representation of a Lie group. The framework also allows us to give a simple derivation of Smale's criterion for relative equilibria. We apply our scheme to various systems, including rotationally invariant systems, the rigid body, fluid flow, and general relativity.     

Symplectic structures associated to Lie-Poisson groups

The Lie-Poisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified, and the corresponding symplectic forms are described. Thus the construction of the Kirillov symplectic form is generalized for Lie-Poisson groups.Supported in part by a Soros Foundation Grant awarded by the American Physical SocietyUnité Associée au C.N.R.S., URA 280     

Algebraic deformation program on minimal nilpotent orbit

We construct an algebraic star product on the minimal nilpotent coadjoint orbit of a simple complex Lie group with a Lie algebra which is not of typeA _n. According to the deformation program, we study the representations of the Lie algebra associated to this orbit.     

All local classical symmetries in Hamiltonian mechanics

Using the formalism of symplectic group actions and coadjoint orbits, we give a complete list of all classical simple Lie algebras which are local symmetries for a given Hamiltonian vector field.     

Uncertainty principles for magnetic structures on certain coadjoint orbits

By building on our earlier work, we establish uncertainty principles in terms of Heisenberg inequalities and of the ambiguity functions associated with magnetic structures on certain coadjoint orbits of infinite-dimensional Lie groups. These infinite-dimensional Lie groups are semidirect products of nilpotent Lie groups and invariant function spaces thereon. The recently developed magnetic Weyl calculus is recovered in the special case of function spaces on abelian Lie groups.     

On tangential star products for the coadjoint Poisson structure

We derive necessary conditions on a Lie algebra from the existence of a star product on a neighbourhood of the origin in the dual of the Lie algebra for the coadjoint Poisson structure which is both differential and tangential to all the coadjoint orbits. In particular we show that when the Lie algebra is semisimple there are no differential and tangential star products on any neighbourhood of the origin in the dual of its Lie algebra.Research partially supported by EC contract CHRX-CT920050     

Dynamical algebras in classical mechanics

For a spectrum-generating algebra of classical observables, it is proven that the phase space dynamics simplifies to a Hamiltonian system on submanifolds of the algebra's dual. These submanifolds are coadjoint orbits if the algebra arises from a symplectic group action. If the Hamiltonian splits into the sum of a function of the algebra generators plus a commuting part, then the dynamics transfers to the dual space and an explicit formula is given for the flow vector field on the coadjoint orbits. A unique feature of the presentation is that all constructions are at the Lie algebra level.     

Quantization of Equivariant Vector Bundles

The quantization of vector bundles is defined. Examples are constructed for the well controlled case of equivariant vector bundles over compact coadjoint orbits. (A coadjoint orbit is a symplectic manifold with a transitive, semisimple symmetry group.) In preparation for the main result, the quantization of coadjoint orbits is discussed in detail. This subject should not be confused with the quantization of the total space of a vector bundle such as the cotangent bundle. Received: 27 February 1998 / Accepted: 5 November 1998     

Quantum orbits of theR-matrix type

Given a simple Lie algebra g, we consider the orbits in g^* which are of theR-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-calledR-matrix bracket. We call an algebra quantizing the latter bracket a quantum orbit of theR-matrix type. We describe some orbits of this type explicitly and we construct a quantization of the whole Poisson pencil on these orbits in a similar way. The notions ofq-deformed Lie brackets, braided coadjoint vector fields, and tangent vector fields are discussed as well.     

On sufficient and necessary conditions for linearity of the transverse Poisson structure

We study the possibility of bringing the transverse Poisson structure to a coadjoint orbit (on the dual of a real Lie algebra) to a normal linear form. We study the relation between two sufficient conditions for linearity of such structures (P. Molino’s condition and our own). We then use these conditions to conclude that, if the isotropy subgroup of the (singular) point in question is compact, or if the isotropy subalgebra is semisimple, then there is a linear transverse Poisson structure to the corresponding coadjoint orbit.     

Kac-moody lie algebras and soliton equations: III. Stationary equations associated with A1(1)

The Hamiltonian structure of stationary soliton equations associated with the AKNS eigenvalue problem is derived in two ways. First, it is shown to arise from the Kostant-Kirillov symplectic structure on a coadjoint orbit in an infinite-dimensional Lie algebra. Second, it is obtained as the restriction to a finite-dimensional manifold of the infinite-dimensional Hamiltonian structure associated with a certain eigenvalue problem polynomial in the eigenvalue parameter.     

Characterization of Lie groups on the cotangent bundle of a Lie group

Characterization, in differential geometric terms, of the groups which can be interpreted as semidirect products of a Lie group G by the group of translations of the dual space of its Lie algebra. Study of the canonical cotangent group of G corresponding to the coadjoint representation. Applications.     

Quantum Surfaces,Special Functions,and the Tunneling Effect

The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible representations of associative algebras and the corresponding trace formulas over leaves with complex polarization are obtained. The noncommutative product on the leaves incorporates a closed 2-form and a measure which (in general) are different from the classical symplectic form and the Liouville measure. The quantum objects are related to some generalized special functions. The difference between classical and quantum geometrical structures could even occur to be exponentially small with respect to the deformation parameter. This is interpreted as a tunneling effect in the quantum geometry.     

Unitary Representations of Nilpotent Super Lie Groups

We show that irreducible unitary representations of nilpotent super Lie groups can be obtained by induction from a distinguished class of sub super Lie groups. These sub super Lie groups are natural analogues of polarizing subgroups that appear in classical Kirillov theory. We obtain a concrete geometric parametrization of irreducible unitary representations by nonnegative definite coadjoint orbits. As an application, we prove an analytic generalization of the Stone-von Neumann theorem for Heisenberg-Clifford super Lie groups.     

Hamiltonian Structures on Coadjoint Orbits of Semidirect Product G = Diff+(S1) ? < C∞(S1, ?)

We consider the semidirect product of diffeomorphisms of the circle D=Diff₊(S ¹) and C (S ¹, ) functions, classify its coadjoint orbits and treat dynamical systems, related to corresponding Lie algebra centrally extended by the Kac-Moody, Virasoro and semidirect product cocycles with arbitrary coefficients. The three-Hamiltonian (in the case of the generalized DWW-type models) and bi-Hamiltonian (for KdV-type models) structures are found and used in the proof of their complete integrability.     

Coadjoint Poisson Actions of Poisson-Lie Groups

Abstract

A Poisson-Lie group acting by the coadjoint action on the dual of its Lie algebra induces on it a non-trivial class of quadratic Poisson structures extending the linear Poisson bracket on the coadjoint orbits.     

Déformations covariantes sur les orbites polarisées d''un groupe de Lie

We prove the existence of covariant star-products on the orbits of the coadjoint representation of a Lie group which admits polarisations.     

Invariant star products and representations of compact semisimple Lie groups

Starting from the work by F. A. Berezin, and earlier paper by the author defined an invariant star product on every nonexceptional Kähler symmetric space. In this Letter a recursion formula is obtained to calculate the corresponding invariant Hochschild 2-cochains for spaces of types II and III. An invariant star product is defined on every integral symplectic (Kähler) homogeneous space of simply-connected compact Lie groups (on every integral orbit of the coadjoint representation). The invariant 2-cochains are obtained from the Bochner-Calabi function of the space. The leading term of the lth-2-cochain is determined by the l-power of the Laplace operator.