Some remarks on the transverse poisson structures of coadjoint orbits

In this paper, we describe how to compute the transverse Poisson structures of coadjoint orbits using Dirac's constraint bracket formula, and we prove that if the isotropy algebra admits a complementary subalgebra, then the transverse structure is, at most, quadratic.     

Linearity of the Transverse Poisson Structure to a Coadjoint Orbit

We prove a simple formula for the transverse Poisson structure to a coadjoint orbit (in the dual of a Lie algebra ) and use it in examples such as and . We also give a sufficient condition on the isotropy subalgebra of so that the transverse Poisson structureto the coadjoint orbit of is linear.     

Semidirect products and the Pukanszky condition

We study the general geometrical structure of the coadjoint orbits of a semidirect product formed by a Lie group and a representation of this group on a vector space. The use of symplectic induction methods gives new insight into the structure of these orbits. In fact, each coadjoint orbit of such a group is obtained by symplectic induction on some coadjoint orbit of a “smaller” Lie group. We study also a special class of polarizations related to a semidirect product and the validity of Pukanszky's condition for these polarizations. Some examples of physical interest are discussed using the previous methods     

The higher order Hamiltonian structures for the modified classical Yang-Baxter equation

We consider constructing the higher order Hamiltonian structures on the dual of the Lie algebra from the first Hamiltonian structure of the coadjoint orbit method. For this purpose we show that the structure of the Lie algebrag is inherited to the algebra of vector fields ong ^* through the solution of the Modified Classical Yang-Baxter equation (Classicalr matrix). We study the algebra that generates the compatible Poisson brackets.This work was supported by Grant Aid for Scientific Research, the Ministry of Education.     

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.     

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     

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 tangential properties of the Gutt 1-product

We establish necessary and sufficient conditions on a Lie algebra g, under which the Gutt 1-product on g¹ is tangential to a given coadjoint orbit.     

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.     

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.     

W-algebras and the equivalence of bihamiltonian,Drinfeld–Sokolov and Dirac reductions

We prove that the classical $W$-algebra associated to a nilpotent orbit in a simple Lie-algebra can be constructed by preforming bihamiltonian, Drinfeld–Sokolov or Dirac reductions. We conclude that the classical $W$-algebra depends only on the nilpotent orbit but not on the choice of a good grading or an isotropic subspace. In addition, using this result we prove again that the transverse Poisson structure to a nilpotent orbit is polynomial and we better clarify the relation between classical and finite $W$-algebras.     

Bogomol'nyi solitons and Hermitian symmetric spaces

We apply the coadjoint orbit method to construct relativistic nonlinear sigma models (NLSM) on the target space of coadjoint orbits coupled with the Chern-Simons (CS) gauge field and we study self-dual solitons. When the target space is given by a Hermitian symmetric space (HSS), we find that the system admits self-dual solitons whose energy is Bogomol'nyi bounded from below by a topological charge. The Bogomol'nyi potential on the Hermitian symmetric space is obtained in the case when the maximal torus subgroup is gauged, and the self-dual equation in the CP(N − 1) case is explored. We also discuss the self-dual solitons in the case of noncompact SU(1, 1) and present a detailed analysis for the rotationally symmetric solutions.     

Symplectic manifolds,coadjoint orbits,and mean field theory

Mean field theory is given a geometrical interpretation as a Hamiltonian dynamical system. The Hartree-Fock phase space is the Grassmann manifold, a symplectic submanifold of the projective space of the full many-fermion Hilbert space. The integral curves of the Hartree-Fock vector field are the time-dependent Hartree-Fock solutions, while the critical points of the energy function are the time-independent states. The mean field theory is generalized beyond determinants to coadjoint orbit spaces of the unitary group; the Grassmann variety is the minimal coadjoint orbit.     

On Quantizing Nonnilpotent Coadjoint Orbits of Semisimple Lie Groups

We prove that there is no consistent polynomial quantization of the coordinate ring of a nonnilpotent coadjoint orbit of a semisimple Lie group.     

Reduction of Poisson algebras at nonzero momentum values

A reduction of a Poisson manifold using the ideal I(J) generated by the momentum map was introduced by Śniatycki and Weinstein (1983). This reduction has been extended to nonzero momentum values μ by two methods: by shifting to zero momentum on a larger space, the product with the coadjoint orbit; and by the method of Wilbour and Kimura (1991, 1993) using the modified ideal I(J − μ). It is shown that these two methods produce isomorphic reduced algebras under the assumptions that the symmetry group is connected and that the stabilizer group of μ also is connected. If the latter assumption fails, the shifted reduced algebra is isomorphic to a (possibly proper) subalgebra of the Wilbour-Kimura algebra.     

Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups

Let ? be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, g, with the Lie algebra ?. We study one and two parameter quantizations ? _h and ? _t,h of ? such that the multiplication on the quantized algebra is invariant under action of the Drinfeld–Jimbo quantum group, U _{h (?)}. In particular, the algebra ? _t,h specializes at h= 0 to a U(?)-invariant ($G$-invariant) quantization, %Ascr; _{t ,0}. We prove that the Poisson bracket corresponding to ? _h must be the sum of the so-called r-matrix and an invariant bracket. We classify such brackets for all semisimple orbits, M, and show that they form a dim H ²(M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, $? _t,h , corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on M. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases. Received: 15 August 1998 / Accepted: 13 January 1999     

Coxeter’s frieze patterns and discretization of the Virasoro orbit

We show that the space of classical Coxeter’s frieze patterns can be viewed as a discrete version of a coadjoint orbit of the Virasoro algebra. The canonical (cluster) (pre)symplectic form on the space of frieze patterns is a discretization of Kirillov’s symplectic form. We relate a continuous version of frieze patterns to conformal metrics of constant curvature in dimension 2.     

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.     

Particles with internal structure: The geometry of classical motions and conservation laws

Classical models of elementary particles endowed with internal structure are constructed using the coadjoint orbit method. The geometry of minimal coupling is introduced and the Wong equations are recovered. Conservation laws are derived in the case of symmetric external Einstein-Yang-Mills fields. As a non-trivial example, classical motions in a B.P.S.T. instanton's field are investigated.     

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