Poisson–Lie Diffeomorphism Groups

We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of 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 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 ⁿ , thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff( ^m ) × FDiff ⁿ ) gives rise to classes of compatible Poisson structures on the space J ( ^m , ⁿ ) of infinite jets of smooth maps ^{m 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.     

Poisson–Lie T-Duality¶for Quasitriangular Lie Bialgebras

We introduce a new 2-parameter family of sigma models exhibiting Poisson–Lie T-duality on a quasitriangular Poisson–Lie group G. The models contain previously known models as well as a new 1-parameter line of models having the novel feature that the Lagrangian takes the simple form , where the generalised metric E is constant (not dependent on the field u as in previous models). We characterise these models in terms of a global conserved G-invariance. The models on G=SU ₂ and its dual G ^* are computed explicitly. The general theory of Poisson–Lie T-duality is also extended, notably the reduction of the Hamiltonian formulation to constant loops as integrable motion on the group manifold. The approach also points in principle to the extension of T-duality in the Hamiltonian formulation to group factorisations D=G⋈M, where the subgroups need not be dual or connected to the Drinfeld double. Received: 22 August 1999 / Accepted: 4 February 2000     

Quadratic Deformations of Lie–Poisson Structures

In this letter, first we give a decomposition for any Lie–Poisson structure associated to the modular vector. In particular, splits into two compatible Lie–Poisson structures if . As an application, we classified quadratic deformations of Lie– Poisson structures on up to linear diffeomorphisms. Research partially supported by NSF of China and the Research Project of “Nonlinear Science”.     

Kontsevich and Takhtajan Construction of Star Product on the Poisson–Lie Group GL(2)

Comparing the star product defined by Takhtajan on the Poisson–Lie group GL(2) and any star product calculated from the Kontsevich's graphs (any K-star product) on the same group, we show, by direct computation, that the Takhtajan star product on GL(2) can't be written as a K-star product.     

Poisson–Lie Generalization of the Kazhdan–Kostant–Sternberg Reduction

The trigonometric Ruijsenaars–Schneider model is derived by symplectic reduction of Poisson–Lie symmetric free motion on the group U(n). The commuting flows of the model are effortlessly obtained by reducing canonical free flows on the Heisenberg double of U(n). The free flows are associated with a very simple Lax matrix, which is shown to yield the Ruijsenaars–Schneider Lax matrix upon reduction.     

One loop renormalizability of the Poisson–Lie sigma models

We present the proof of the one loop renormalizability in the strict field theoretic sense of the Poisson–Lie σ-models. The result is valid for any Drinfeld double and it relies solely on the Poisson–Lie structure encoded in the target manifold.     

Orbifold compactifications with three families of Su(3) ×Su(2) ×U(1)n

We construct several N = 1 supersymmetric three-generation models with SU(3)×SU(2)×U(1)ⁿ gauge symmetry, obtained from orbifold compactification of the heterotic string in the presence of constant gauge-background fields. This Wilson-line mechanism also allows us to eliminate extra colour triplets which could mediate fast proton decay.     

Dissipation in Lie–Poisson systems and the Lorenz-84 model

The relation between dissipation and the symplectic structure of the momentum-space is studied in so(3) Lie algebra and in 2D fluid dynamics. Three kinds of dissipative mechanisms are discussed and a general bracket formalism is introduced. A chaotic dynamical system due to Lorenz, and largely studied in low-dimensional models of geophysical fluid dynamics, is analysed in its geometric and dynamical features, by means of the formalism previously introduced. A mechanism of energy transfer for this low-order model is discussed.     

The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

The Lie–Rinehart algebra of a (connected) manifold ${\mathcal {M}}$ , defined by the Lie structure of the vector fields, their action and their module structure over ${C^\infty({\mathcal {M}})}$ , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra ${\Lambda_{R}({\mathcal {M}})}$ , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact ${{\mathcal {M}}}$ ) ${Z}$ which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z  =  i z, z  =  0 and ${z = \hbar}$ , respectively; canonical quantization uniquely follows from such a general geometrical structure. For ${z =\hbar \neq 0}$ , the regular factorial Hilbert space representations of ${\Lambda_{R}({\mathcal{M}})}$ describe quantum mechanics on ${{\mathcal {M}}}$ . For z  =  0, if Diff( ${{\mathcal {M}}}$ ) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on ${{\mathcal {M}}}$ .     

Twisted Poincaré Duality for some Quadratic Poisson Algebras

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier. This research of the first author was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme. The second author was supported by EPSRC Grant EP/D034167/1.     

Ricci flow,Courant algebroids,and renormalization of Poisson–Lie T-duality

We use a generalized Ricci tensor, defined for generalized metrics in Courant algebroids, to show that Poisson–Lie T-duality is compatible with the 1-loop renormalization group flow.     

Poisson Cohomology of SU (2)-Covariant “Necklace” Poisson Structures on S 2

Abstract

We compute the Poisson cohomology of the one-parameter family of SU(2)-covariant Poisson structures on the homogeneous space S ²=?P ¹=SU(2)/U(1), where SU(2) is endowed with its standard Poisson–Lie group structure, thus extending the result of Ginzburg [2] on the Bruhat–Poisson structure which is a member of this family. In particular, we compute several invariants of these structures, such as the modular class and the Liouville class. As a corollary of our computation, we deduce that these structures are nontrivial deformations of each other in the direction of the standard rotation-invariant symplectic structure on S ²; another corollary is that these structures do not admit smooth rescaling.     

The Lie–Poisson representation of the nonlinearized eigenvalue problem of the Kac–van Moerbeke hierarchy

The Kac–van Moerbeke hierarchy is studied by a 3×3 discrete eigenvalue problem and the corresponding nonlinearized one an integrable Poisson map with a Lie–Poisson structure is also presented. Moreover, the 2×2 nonlinearized eigenvalue problem associated with the Kac–van Moerbeke hierarchy is proved to be a reduction of the Poisson map on the leaves of the symplectic foliation.     

Poisson Reduction and Branes in Poisson–Sigma Models

We analyse the problem of boundary conditions for the Poisson–Sigma model and extend previous results showing that non-coisotropic branes are allowed. We discuss the canonical reduction of a Poisson structure to a submanifold, leading to a Poisson algebra that generalizes Diracs construction. The phase space of the model on the strip is related to the (generalized) Dirac bracket on the branes through a dual pair structure.Mathematics Subject Classifications (2000). 81T45, 53D17, 81T30, 53D55.     

活动标架系下O(n)非线性σ模型Laxpair矩阵的Poisson—Lie结构

在活动标架系下给出了O(n)/O(n-1)对称空间上非线性σ模型的Poisson-Lie括号,并用协变分解的方法讨论了活动标架与固定标架系的协交关系,得到的r、s矩阵可以明显地看出其依赖于场量的部分是由于S^n－1流形上的联络引起的.