Lie-Poisson Structure Research Articles

A Poisson^Lie Framework for Rational Reductions of the KP Hierarchy

We give a simple proof of I. Krichever's theorem on rational reductions of the Kadomtsev–Petviashvili hierarchy by using the Poisson–Lie structure on the group of pseudo-differential symbols.

Poisson–Lie Diffeomorphism Groups

We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of ℝ n . Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra W n of formal vector fields on ℝ n . We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson–Lie diffeomorphism groups gives rise to large classes of compatible Poisson structures on ℝ n , thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff(ℝ m ) × FDiffℝ n ) gives rise to classes of compatible Poisson structures on the space J ∞(ℝ m ,ℝ n ) of infinite jets of smooth maps ℝ m → ℝ n , which makes it also a Poisson space for this action. Poisson modules of generalized densities are also constructed. Initial steps towards a classification of these structures are taken.

Symmetry reduction of discrete Lagrangian mechanics on Lie groups

For a discrete mechanical system on a Lie group G determined by a (reduced) Lagrangian l , we define a Poisson structure via the pull-back of the Lie–Poisson structure on the dual of the Lie algebra g * by the corresponding Legendre transform. The main result shown in this paper is that this structure coincides with the reduction under the symmetry group G of the canonical discrete Lagrange 2-form ω L on G× G. Its symplectic leaves then become dynamically invariant manifolds for the reduced discrete system. Links between our approach and that of groupoids and algebroids as well as the reduced Hamilton–Jacobi equation are made. The rigid body is discussed as an example.

The nonlinearized eigenvalue problem of the Toda hierarchy in the Lie–Poisson framework

A 3×3 discrete eigenvalue problem associated with Toda hierarchy is presented. After the nonlinearization procedure, the 3×3 discrete eigenvalue problem is turned into an integrable Poisson map on the Poisson manifold R 3 N with a Lie–Poisson structure. As a reduction of the Lie–Poisson structure on the co-adjoint orbit, the standard symplectic structure on the symplectic manifold R 2 N is obtained. The Poisson map restricted on the leaves of the symplectic foliation is reduced to a usual symplectic map, which is exactly the nonlinearized 2×2 eigenvalue problem.

On the Lie–Poisson structure of the nonlinearized discrete eigenvalue problem

A 3×3 discrete eigenvalue problem and corresponding discrete soliton equations are proposed. Under a constraint between the potentials and eigenfunctions, the 3×3 discrete eigenvalue problem is nonlinearized into an integrable Poisson map with a Lie–Poisson structure. Further, a reduction of the Lie–Poisson structure on the co-adjoint orbit yields the standard symplectic structure. The Poisson map is reduced to the usual symplectic map when it is restricted on the leaves of the symplectic foliation.

Lie–Poisson structure for the homogeneous motion of self-gravitating compressible fluids

In this paper we apply the theory developed by Marsden, Ratiu, and Weinstein for the reduction of a Hamiltonian system defined on the cotangent bundle of a Lie group to a Hamiltonian system in the coalgebra of a semidirect product to study the motion of a self-gravitating homogeneous compressible ideal fluid with a variable ellipsoidal boundary, assuming that the motions are given by invertible linear transformations. The relation between the Lie–Poisson equations obtained and the classical Dyson equations is discussed, and the Hamiltonian structure for the homogeneous expansion of a free nonrotating ellipsoid is derived.

Completely Integrable Systems Connected with Lie Algebras

In a recent paper Ballersteros and Ragnisco (1998) have proposed a new method of constructing integrable Hamiltonian systems. A new class of integrable systems may be devised using the following sequence: \(\mathcal{A} \to \Lambda \to C \to \tilde \Lambda \to \{ .,\;.\} _{\tilde \Lambda } \to (A,\;\vartriangle )\), where A is a Lie algebra \({\text{(R}}^{\text{3}} ,[.,\;.]),\;\Lambda \) is a Lie–Poisson structure on R3, C is a Casimir for \(\Lambda ,\;\{ .,\;.\} _{\tilde \Lambda } \) is a reduced Poisson bracket and (A, ▵) is a bialgebra. We study the relation between a Lie-Poisson stucture Λ and a reduced Poisson bracket \(\{ ,\;\} _{\tilde \Lambda } \), which is a key element in using the Lie algebra A to constructing this sequence. New examples of Lie algebras and their related integrable Hamiltonian systems are given.

Factorization Dynamics and Coxeter--Toda Lattices

It is shown that the factorization relation on simple Lie groups with standard Poisson Lie structure restricted to Coxeter symplectic leaves gives an integrable dynamical system. This system can be regarded as a discretization of the Toda flow. In case of SL n the integrals of the factorization dynamics are integrals of the relativistic Toda system. A substantial part of the paper is devoted to the description of symplectic leaves in simple complex Lie groups, its Borel subgroups and their doubles.

Pseudodifferential operators on differential groupoids

We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the PoincareBirkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual A∗ of a Lie algebroid.

Lie Groupoid C * -Algebras and Weyl Quantization

For any Lie groupoid $G$, the vector bundle $g^*$ dual to the associated Lie algebroid $g$ is canonically a Poisson manifold. The (reduced) C*-algebra of $G$ (as defined by A. Connes) is shown to be a strict quantization (in the sense of M. Rieffel) of $g^*$. This is proved using a generalization of Weyl's quantization prescription on flat space. Many other known strict quantizations are a special case of this procedure; on a Riemannian manifold, one recovers Connes' tangent groupoid as well as a recent generalization of Weyl's prescription. When $G$ is the gauge groupoid of a principal bundle one is led to the Weyl quantization of a particle moving in an external Yang-Mills field. In case that $G$ is a Lie group (with Lie algebra $g$) one recovers Rieffel's quantization of the Lie-Poisson structure on $g^*$. A transformation group C*-algebra defined by a smooth action of a Lie group on a manifold $Q$ turns out to be the quantization of the semidirect product Poisson manifold $g^*x Q$ defined by this action.

Poisson–Lie structures on infinite-dimensional jet groups and quantum groups related to them

We study the problem of classifying all Poisson–Lie structures on the group G∞ of formal diffeomorphisms of the real line R1 which leave the origin fixed, as well as the extended group of diffeomorphisms G0∞⊃G∞ whose action on R1 does not necessarily fix the origin. A complete local classification of all Poisson–Lie structures on the groups G∞ and G0∞ is given. This includes a classification of all Lie-bialgebra structures on the Lie algebra G∞ of G∞, which we prove to be all of the coboundary type, and a classification of all Lie-bialgebra strucutures on the Lie algebra G0∞ (the Witt algebra) of G0∞ which also turned out to be all of the coboundary type. A large class of Poisson structures on the space Vλ of λ-densities on the real line is found such that Vλ becomes a homogeneous Poisson space under the action of the Poisson–Lie group G∞. We construct a series of quantum semigroups whose quasiclassical limits are finite-dimensional Poisson–Lie quotient groups of G∞ and G0∞.

Geometric phases, reduction and Lie-Poisson structure for the resonant three-wave interaction

Hamiltonian Lie-Poisson structures of the three-wave equations associated with the Lie algebras SU (3) and SU (2, 1) are derived and shown to be compatible. Poisson reduction is performed using the method of invariants, and geometric phases associated with the reconstruction are calculated. These results can be applied to applications of nonlinear-waves in, for instance, nonlinear optics. Some of the general structures presented in the latter part of this paper are implicit in the literature; our purpose is to put the three-wave interaction in the modern setting of geometric mechanics and to explore some new things, such as explicit geometric phase formulas, as well as some old things, such as integrability, in this context.

Poisson-Lie structures on Poincaré and Euclidean groups in three dimensions

The complete list of Poisson-Lie structures on Poincaré and Euclidean groups in three dimensions is presented. Some new solutions for inhomogenous are given.

Drinfeld-Sokolov Reduction for Difference Operators and Deformations of ${\cal W}$ -Algebras¶I. The Case of Virasoro Algebra

We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical W-algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL(2). The nontrivial consistency conditions fix the choice of the classical r-matrix defining the Poisson-Lie structure on the loop group LSL(2), and this leads to a new elliptic classical r-matrix. The reduced Poisson algebra coincides with the deformation of the classical Virasoro algebra previously defined in q-alg/9505025. We also consider a discrete analogue of this Poisson algebra. In the second part (q-alg/9702016) the construction is generalized to the case of an arbitrary semisimple Lie algebra.

Twisted Lie group $C^*$-algebras as strict quantizations

A nonzero 2-cocycle Γ∈ Z2(g, R) on the Lie algebra g of a compact Lie group G defines a twisted version of the Lie–Poisson structure on the dual Lie algebra g*, leading to a Poisson algebra C∞ (g*(Γ)). Similarly, a multiplier c∈ Z2(G, U(1)) on G which is smooth near the identity defines a twist in the convolution product on G, encoded by the twisted group C-algebra C*(G,c). Further to some superficial yet enlightening analogies between C∞ (g*(Γ)) and C*(G,c), it is shown that the latter is a strict quantization of the former, where Planck’s constant ħ assumes values in (Z\{0})-1. This means that there exists a continuous field of C*-algebras, indexed by ħ ∈ 0 ∪ (Z\{0})-1, for which A0= C0(g*) and Aħ=C*(G,c) for ħ ≠ 0, along with a cross-section of the field satisfying Dirac’s condition asymptotically relating the commutator in Aħ to the Poisson bracket on C∞(g*(Γ)). Note that the ‘quantization’ of ħ does not occur for Γ=0.

Linearization of some Poisson-Lie tensor

In this paper we give some example of linear, linearizable and non-linearizable Poisson-Lie structures. We show that any Poisson-Lie tensor P on a Lie group G such that g ∗ is reductive is analytically linearizable, this property being not always satisfied by a general Poisson tensor. We also give examples of linearizable and non-linearizable exact Poisson-Lie structures on some nilpotent Lie groups.

Stability of a bottom-heavy underwater vehicle

We study stability of underwater vehicle dynamics for a six-degree-of-freedom vehicle modeled as a neutrally buoyant, submerged rigid body in an ideal fluid. We consider the case in which the center of gravity and the center of buoyancy of the vehicle are noncoincident such that gravity introduces an orientation-dependent moment. Noting that Kirchhoff's equations of motion for a submerged rigid body are Hamiltonian with respect to a Lie-Poisson structure, we derive the Lie-Poisson structure for the underwater vehicle dynamics with noncoincident centers of gravity and buoyancy. Using the energy-Casimir method, we then derive conditions for Lyapunov stability of relative equilibria, i.e. stability of motions corresponding to constant translations and rotations. The conditions reveal for the vehicle stability problem the relevant design parameters, which in some cases can be interpreted as control parameters. Further, the formulation provides a setting for exploring the stabilizing and destabilizing effects of dissipation and externally applied control forces and torques.

Quadratic Poisson brackets and the Drinfeld theory for associative algebras

The paper is devoted to the Poisson brackets compatible with multiplication in associative algebras. These brackets are shown to be quadratic and their relations with the classical Yang--Baxter equation are revealed. The paper also contains a description of Poisson Lie structures on Lie groups whose Lie algebras are adjacent to an associative structure.

A note on the symplectic structure on the dressing group in the sinh-Gordon model

We analyze the symplectic structure on the dressing group in the sinh-Gordon model by calculating explicitly the Poisson bracket { g ⊗, g} where g is the dressing group element which creates a generic one-soliton solution from the vacuum. Our result is that this bracket does not coincide with the Semenov-Tian-Shansky one. The latter induces a Lie-Poisson structure on the dressing group. To get the bracket obtained by us from the Semenov-Tian-Shansky bracket we apply the formalism of the constrained Hamiltonian systems. The constraints on the dressing group appear since the element which generates one-soliton solutions from the vacuum has a specific form.

Graded poisson-lie structures on general linear groups

The external algebra on GL(N) is equipped with graded Poisson brackets compatible with the group action. We prove that there exist two and only two corresponding Poisson-Hopf superalgebras on M and obtain their explicit descriptions. We show that precisely these superalgebras arise in the semiclassical limit of bicovariant differential calculi on the quantum linear group GLq(N).