Poisson Reduction Research Articles

Poisson-Lie group of pseudodifferential symbols

We introduce a Lie bialgebra structure on the central extension of the Lie algebra of differential operators on the line and the circle (with scalar or matrix coefficients). This defines a Poisson--Lie structure on the dual group of pseudodifferential symbols of an arbitrary real (or complex) order. We show that the usual (second) Benney, KdV (or GL_n--Adler--Gelfand--Dickey) and KP Poisson structures are naturally realized as restrictions of this Poisson structure to submanifolds of this ``universal'' Poisson--Lie group. Moreover, the reduced (=SL_n) versions of these manifolds (W_n-algebras in physical terminology) can be viewed as subspaces of the quotient (or Poisson reduction) of this Poisson--Lie group by the dressing action of the group of functions. Finally, we define an infinite set of functions in involution on the Poisson--Lie group that give the standard families of Hamiltonians when restricted to the submanifolds mentioned above. The Poisson structure and Hamiltonians on the whole group interpolate between the Poisson structures and Hamiltonians of Benney, KP and KdV flows. We also discuss the geometrical meaning of W_\infty as a limit of Poisson algebras W_\epsilon as \epsilon goes to 0.

Quantization and representation theory of finiteW algebras

In this paper we study the finitely generated algebras underlying $W$ algebras. These so called 'finite $W$ algebras' are constructed as Poisson reductions of Kirillov Poisson structures on simple Lie algebras. The inequivalent reductions are labeled by the inequivalent embeddings of $sl_2$ into the simple Lie algebra in question. For arbitrary embeddings a coordinate free formula for the reduced Poisson structure is derived. We also prove that any finite $W$ algebra can be embedded into the Kirillov Poisson algebra of a (semi)simple Lie algebra (generalized Miura map). Furthermore it is shown that generalized finite Toda systems are reductions of a system describing a free particle moving on a group manifold and that they have finite $W$ symmetry. In the second part we BRST quantize the finite $W$ algebras. The BRST cohomology is calculated using a spectral sequence (which is different from the one used by Feigin and Frenkel). This allows us to quantize all finite $W$ algebras in one stroke. Explicit results for $sl_3$ and $sl_4$ are given. In the last part of the paper we study the representation theory of finite $W$ algebras. It is shown, using a quantum version of the generalized Miura transformation, that the representations of finite $W$ algebras can be constructed from the representations of a certain Lie subalgebra of the original simple Lie algebra. As a byproduct of this we are able to construct the Fock realizations of arbitrary finite $W$ algebras.

Construction of KP hierarchies in terms of finite number of fields and their abelianization

The 2 M-boson representations of the KP hierarchy are constructed in terms of M mutually independent two-boson KP representations for arbitrary number M. Our construction establishes the multi-boson representations of the KP hierarchy as consistent Poisson reductions of the standard KP hierarchy within the R-matrix scheme. As a byproduct we obtain a complete description of any finitely-many-field formulation of the KP hierarchy in terms of Darboux coordinates with respect to the first Hamiltonian structure. This results in a series of representations of W 1+∞ algebra made out of an arbitrary even number of boson fields.

Poisson Manifolds Associated with Group Actions and Classical Triangular r-Matrices

Let P be a Poisson G-space and Λ a classical triangular r-matrix. Using the Poisson reduction, we construct a new Poisson structure P Λ on P. For this new Poisson structure P Λ , we construct its symplectic groupoid, describe its symplectic leaves, and classify its symplectic realizations. The deformation quantization of P Λ is also discussed.

Hamiltonian structures and stability for rigid bodies with flexible attachments

The dynamics of a rigid body with flexible attachments is studied. A general framework for problems of this type is established in the context of Poisson manifolds and reduction. A simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration. Second, the Energy-Casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out. The Poisson structure and stability results take into account vibrations of the string, rotations of the rigid body, their coupling at the point of attachment, and centrifugal and Coriolis forces.

Poisson reduction and quantization for the n+1 photon

For a dynamical system in which the constraints are given by the vanishing of a singular momentum map J, reduction in the usual group-theoretic sense may not be possible. Nonetheless, one may still ‘‘reduce’’ J−1(0), at least on the level of Poisson algebras. An example of such a singular constrained system is the ‘‘n+1 photon,’’ that is, a massless, spinless particle in (n+1)-dimensional Minkowski space-time. We apply the generalized reduction procedure to the n+1 photon, explicitly constructing the Poisson algebra of gauge invariant observables. This technique also enables us to completely analyze the effects of the singularities in J−1(0) on the system. We then quantize, obtaining results which are in agreement with a quantization of the extended phase space and the subsequent imposition of the constraint.