Dynamical algebras for Pöschl-Teller Hamiltonian hierarchies

The dynamical algebras of the trigonometric and hyperbolic symmetric Pöschl-Teller Hamiltonian hierarchies are obtained. A kind of discrete-differential realizations of these algebras are found which are isomorphic to so(3, 2) Lie algebras. In order to get them, first the relation between ladder and factor operators is investigated. In particular, the action of the ladder operators on normalized eigenfunctions is found explicitly. Then, the whole dynamical algebras are generated in a straightforward way.     

Nonlinear dynamical systems,first integrals,Bose operators,and Lie algebras

A connection between nonlinear autonomous systems of ordinary differential equations, first integrals, Bose operators and Lie algebras is described. An extension to nonlinear partial differential equations is given.     

Integrable systems on so(n)-valued Lie algebras on higher genus curves

We consider special quasi-graded so(n)-valued Lie algebras on higher genus curves. Using them we find new Lie theoretical interpretation of the integrability of generalized Manakov tops and generalized Clebsch systems and explicitly construct their isomorphism on trajectories.     

Noncommutative *-product continual Lie algebras and solvable models

Noncommutative counterparts of exactly solvable models are proposed on the basis of *-product continual Lie algebras. Examples of noncommutative Liouville and sine/sh-Gordon equations are discussed.     

黑克、布劳、及伯曼-温采尔代数的诱导及分导系数和量子经典李代数的耦合与重新耦合系数

本文总结了计算黑克、布劳、及伯曼 温采尔代数在各种工数链下诱导及分导系数的线性方程方法(LEM)。特别强调了关于A,B,C,D型李代数及其量子情形与其中心代数之间的舒尔 魏尔 布劳双关性关系。这一关系使我们能够利用相应中心代数的诱导及分导系数计算出经典李代数及其量子情形的耦合与重新耦合系数。讨论了从该方法得到B,C,D型李代数不可约表示克罗内克积分解的应用。基于LEM还得到了处理对应于置换群CG系列问题的黑克代数张量积的方法。     

Graded differential lie algebras and model building

We develop a mathematical concept towards gauge field theories based upon a Hilbert space endowed with a representation of a skew-adjoint Lie algebra and an action of a generalized Dirac operator. This concept shares common features with the non-commutative geometry à la Connes/Lott, differs from that, however, by the implementation of skew-adjoint Lie algebras instead of unital associative *-algebras. We present the physical motivation for our approach and sketch its mathematical strategy. Moreover, we comment on the application of our method to the standard model and the flipped SU(5)×U(1)-grand unification model.     

The mermin-wagner phenomenon and cluster properties of one- and two-dimensional systems

We give optimal conditions concerning the range of interactions for the absence of spontaneous breakdown of continuous symmetries for one- and two-dimensional quantum and classical lattice and continuum systems. For a class of models verifying infrared bounds our conditions are necessary and sufficient. Using the same techniques we obtain a priori bounds on clustering for systems with continuous symmetry, improving results of Jasnow and Fisher.Partially supported by CAPES-PICD.Partially supported by the CNP_q.Partially supported by N.S.F. under grant MCS 7801433.     

Polynomial poisson algebras for two-dimensional classical superintegrable systems and polynomial associative algebras for quantum superintegrable systems

The integrals of motion of the classical two-dimensional superintegrable systems close in a restrained polynomial Poisson algebra, whose general form is discussed. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The polynomial Poisson algebra is deformed to a polynomial associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is conjectured that the finite-dimensional representations of the polynomial algebra are determined by the energy eigenvalues of the superintegrable system. The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal for a large number of two-dimensional superintegrable systems. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Investigation of dodecylammonium adsorption on mica,albite and quartz surfaces by QM/MM simulation

The absorption mechanisms of collector and mineral surface structures play important roles in studies of lepidolite flotation. In this work, quantum mechanics (QM) and hybrid quantum mechanics/molecular mechanics (MM) methods were implemented to investigate the flotation mechanisms of lepidolite from muscovite, quartz and albite. The crystal structures, electron density distributions, bonds and the densities of states of lepidolite were calculated and compared with those of muscovite. The adsorption structures and energies of monomer dodecylammonium (DDA) on the three different minerals were also calculated. The headgroup of the DDA cation was found to adsorb on the surface of minerals, with its hydrophobic tail stretching into the vacuum slab, approximately perpendicular to the surface. Simulation results indicate that the purity of fine lepidolite is limited by the existence of muscovite, due to their similarities in surficial structure and properties. Other gangues were found to be removed efficiently with the use of acidic conditions. The results are in good agreement with other experiments. Compared with simple MM simulations, the use of the QM/MM methods to investigate the adsorption on minerals without specific forcefield parameters was concluded to be a more accurate method to attain monomer surfactant–mineral adsorption energies.     

Flux Hamiltonians, Lie algebras, and root lattices with minuscule decorations

We study a family of Hamiltonians of fermions hopping on a set of lattices in the presence of a background gauge field. The lattices are constructed by decorating the root lattices of various Lie algebras with their minuscule representations. The Hamiltonians are, in momentum space, themselves elements of the Lie algebras in these same representations. We describe various interesting aspects of the spectra, which exhibit a family resemblance to the Dirac spectrum, and in many cases are able to relate them to known facts about the relevant Lie algebras. Interestingly, various realizable lattices such as the kagomé and pyrochlore can be given this Lie algebraic interpretation, and the particular flux Hamiltonians arise as mean-field Hamiltonians for spin-1/2 Heisenberg models on these lattices.     

Classical spectrum generating algebra of the Kepler-Coulomb system and action-angle variables

The classical spectrum generating algebra for the one-dimensional Kepler-Coulomb system is computed and a set of two corresponding constants of motion depending explicitly on time is obtained. Such constants supply the solution to the motion in an algebraic way. The connection of the spectrum generating algebra and the action-angle variables of the system is also shown.     

The Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems

This paper investigates the Lie symmetries and Noether conserved quantities of discrete non-conservative mechanical systems. The variational principle of discrete mechanics, from which discrete motion equations of systems are deduced, is generalized to the case of including the time variational. The requirement for an invariant group transformation is defined to be the Lie symmetry and the criterion when the Noether conserved quantities may be obtained from Lie symmetries is also presented. An example is discussed for applications of the results.     

Faux-boolean algebras,classical probability,and determinism

It is possible to define, for any quantum system, an algebra of definite-valued events—those events that are definitely occurrent or non-occurrent. It is shown that two different sets of constraints on the algebra of definite-valued events are each equivalent to the definition of that set as a certain pseudo-Boolean algebra.     

Finite systems,fractional Fourier transforms and their finite phase spaces

Various harmonic oscillator models define — in a sense to be explained here — fractional Fourier transforms (up to a phase). The fractionalization of the Fourier integral transform is well understood; the finite case is less. There are several discrete and finite oscillator models that contract to the continuous, integral model. The Ankara model can be thought as a ring of point masses joined by springs to their equilibrium positions and to each other; the Cuernavaca model uses the su(2) algebra with a distinct physical interpretation. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

Lie algebroid morphisms, Poisson sigma models, and off-shell closed gauge symmetries

Chern–Simons (CS) gauge theories in three dimensions and the Poisson sigma model (PSM) in two dimensions are examples of the same theory, if their field equations are interpreted as morphisms of Lie algebroids and their symmetries (on-shell) as homotopies of such morphisms. We point out that the (off-shell) gauge symmetries of the PSM in the literature are not globally well defined for non-parallelizable Poisson manifolds and propose a covariant definition of the off-shell gauge symmetries as left action of some finite-dimensional Lie algebroid.

Our approach allows us to avoid complications arising in the infinite-dimensional super-geometry of the BV- and AKSZ-formalism. This preprint is a starting point in a series of papers meant to introduce Yang–Mills type gauge theories of Lie algebroids, which include the standard YM theory, gerbes, and the PSM.     

Lie symmetries and exact solutions for a short-wave model

In this paper, the Lie symmetry analysis and generalized symmetry method are performed for a short-wave model. The symmetries for this equation are given, and the phase portraits of the traveling wave systems are analyzed using the bifurcation theory of dynamical systems. The exact parametric representations of four types of traveling wave solutions are obtained.     

Affine Lie algebras, string junctions and 7-branes

We consider the realization of affine ADE Lie algebras as string junctions on mutually non-local 7-branes in Type 1113 string theory. The existence of the affine algebra is signaled by the presence of the imaginary root junction δ, which is realized as a string encircling the 7-brane configuration. The level k of an affine representation partially constrains the asymptotic (p, q) charges of string junctions departing the configuration. The junction intersection form reproduces the full affine inner product, plus terms in the asymptotic charges.     

Formal Deformations,Contractions and Moduli Spaces of Lie Algebras

Jump deformations and contractions of Lie algebras are inverse concepts, but the approaches to their computations are quite different. In this paper, we contrast the two approaches, showing how to compute the jump deformations from the miniversal deformation of a Lie algebra, and thus arrive at the contractions. We also compute contractions directly. We use the moduli spaces of real 3-dimensional and complex 3 and 4-dimensional Lie algebras as models for explaining a deformation theory approach to computation of contractions. The research of the authors was partially supported by grants from the Mathematisches Forschungsinstitut Oberwolfach, OTKA T043641, T043034 and the University of Wisconsin-Eau Claire.