Su (2) Poisson–Lie T Duality

Poisson–Lie target space duality is a framework where duality transformations are properly defined. In this Letter, we investigate the dual pair of -models defined by the double SO(3,1) in the Iwasawa decomposition.     

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.     

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”.     

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–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.     

Integrable Geodesic Flows on the (Super)extension of the Bott–Virasoro Group

In this Letter, we present an answer to the question posed by Marcel, Ovsienko and Roger in their paper (Lett. Math. Phys. 40 (1997), 31–39). The Itô equation, modified dispersive water wave equation and modified dispersionless long wave equation are shown to be the geodesic flows with respect to an L ² metric on the semidirect product space Diff ^s C (S ¹), where Diff ^s (S ¹) is the group of orientation-preserving Sobolev H ^s diffeomorphisms of the circle. We also study the geodesic flows with respect to H ¹ metric. The geodesic flows in this case yield different integrable systems admitting nonlinear dispersion terms. These systems exhibit more general wave phenomena than usual integrable systems. Finally, we study an integrable geodesic flow on the extended Neveu–Schwarz space.     

The Hamiltonian systems on the Poisson structure of the quasi-classical limit of GL q (∞)

We consider the Hamiltonian systems on the Poisson structure of GL() which is introduced from the quantum group GL _q () by the so-called quasi-classical limit of GL _q (). Furthermore, we show that the Toda lattice hierarchy is a Hamiltonian system of this structure.     

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 ${{\mathcal {M}}}$ .     

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.     

Design and Construction of a Three-Color Gold–Copper-Vapor Laser and the Output-Power Dependence on the Frequency and Buffer-gas Pressure

We design and construct a three-color gold–copper-vapor laser emitting green (510.6 nm), yellow (578.2 nm), and red (627.8 nm) light. The maximum measured total average output power is 12 W under sealed-off conditions. We divide the active medium into three zones (two at both ends for copper and the central zone for gold) in order to vaporize both gold and copper simultaneously. For this purpose, we use a single type of thermal insulator to change the temperature along the medium by varying its thickness, which is the main point in our design. In addition, we carry out some experiments to distinguish the dependence of the output power on the frequency and buffer-gas pressure; the measured ratio of these three wavelengths, green : yellow: red, is 22 : 10 : 7.     

Construction of the Dirichlet–Voronoi translation-compatible polyhedrons based on the Zelling parameters

A method of constructing the Dirichlet–Voronoi translation-compatible polyhedrons for sublattices of complex crystal compounds is presented. With the use of the Zelling parameters, this problem is reduced to a linear programming problem which can be solved on a personal computer for a finite ratio of translation-compatible polyhedron volumes. Particular solutions are derived for cubic and tetragonal systems with different orientations of the Bravais frame and polyhedron types.     

Construction of the Γ9 × (g+ 2h) Jahn–Teller Hamiltonian

The coupling coefficients describing the linear Γ ₉× (g+2h) Jahn–Teller Hamiltonian in icosahedral symmetry are derived, and various coupling schemes are discussed.     

Double Product Integrals and Enriquez Quantisation of Lie Bialgebras II: The Quantum Yang–Baxter Equation

For a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra , let be the vector space of tensors over equipped with the Itô Hopf algebra structure derived from the associative multiplication in . It is shown that a necessary and sufficient condition that the double product integral satisfy the quantum Yang–Baxter equation over is that satisfy the same equation over the unital associative algebra got by adjoining a unit element to . In particular, the first-order coefficient r₁ of r[h] satisfies the classical Yang–Baxter equation. Using the fact that the multiplicative inverse of is where is the inverse of in we construct a quantisation of an arbitrary quasitriangular Lie bialgebra structure on in the unital associative subalgebra of consisting of formal power series whose zero order coefficient lies in the space of symmetric tensors. The deformation coproduct acts on by conjugating the undeformed coproduct by and the coboundary structure r of is given by where is the flip.Mathematical Subject Classification (2000). 53D55, 17B62     

Effect of the donor–acceptor properties of ligands on the spectroscopic and electrochemical properties of mixed-ligand complexes of Pt(II) and Ir(III) with cyclometalated 2-phenylbenzothiazole

The results of the spectroscopic NMR (¹H, ¹³C, and ¹⁹⁵Pt), infrared, optical, and voltammetric characteristics of the mixed-ligand complexes of Pt(II) and Ir(III) with metalated 2-phenylbenzothiazole and tert-butylisocyanide (tBuNC), acetonitrile (AN), ethylenediamine (En), O-ethyldithiocarbamate (Exn^–), and diethyldithiocarbamate (Dtc–) ions are presented. It is demonstrated that the change in donor–acceptor interaction of ligands tBuNC, AN, En, Exn^–, and Dtc^– with metal leads to an increase in the energy of the highest occupied molecular orbital of the complexes and is accompanied by a shift of the cathode potential of the metal-centered oxidation, a bathochromic shift of the spin-allowed and spin-forbidden metal-tocyclometalated ligand optical charge transfer transitions, and an increase the degree of mixing of the ¹MLCT and triplet intraligand states, responsible for the phosphorescence of the complexes.     

On The Structure and Representations of the Insertion–Elimination Lie Algebra

We examine the structure of the insertion–elimination Lie algebra on rooted trees introduced in Connes and Kreimer (Ann. Henri Poincar 3(3):411–433, 2002). It possesses a triangular structure , like the Heisenberg, Virasoro, and affine algebras. We show in particular that it is simple, which in turn implies that it has no finite-dimensional representations. We consider a category of lowest-weight representations, and show that irreducible representations are uniquely determined by a “lowest weight” . We show that each irreducible representation is a quotient of a Verma-type object, which is generically irreducible.      

Automorphisms and Derivations of the Insertion–Elimination Algebra and Related Graded Lie Algebras

This paper addresses several structural aspects of the insertion–elimination algebra \({\mathfrak{g}}\), a Lie algebra that can be realized in terms of tree-inserting and tree-eliminating operations on the set of rooted trees. In particular, we determine the finite-dimensional subalgebras of \({\mathfrak{g}}\), the automorphism group of \({\mathfrak{g}}\), the derivation Lie algebra of \({\mathfrak{g}}\), and a generating set. Several results are stated in terms of Lie algebras admitting a triangular decomposition and can be used to reproduce results for the generalized Virasoro algebras.