Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems

The integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic Poisson algebra, whose the general form is investigated. Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system. The quadratic Poisson algebra is deformed to a quantum associative algebra, the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique. It is shown that, the finite dimensional representations of the quadratic 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 all two dimensional superintegrable systems with quadratic integrals of motion.Comment: 28 pages, Late

Finite Dimensional Representations of Quadratic Algebras with Three Generators and Applications

The finite dimensional representations of associative quadratic algebras with three generators are investigated by using a technique based on the deformed parafermionic oscillator algebra. One application on the calculation of the eigenvalues of the two-dimensional superintegrable systems is discussed.Comment: 14 pages, LaTeX 2e, Talk given at the VI International Wigner Symposium, 16-22 August 1999, Instabul, Turke

Parabosonic and parafermionic algebras. Graded structure and Hopf structures

Parabosonic $P_{B}^{(n)}$ and parafermionic $P_{F}^{(n)}$ algebras are described as quotients of the tensor algebras of suitably choosen vector spaces. Their (super-) Lie algebraic structure and consequently their (super-) Hopf structure is shortly discussed. A bosonisation-like construction is presented, which produces an ordinary Hopf algebra $P_{B(K^{\pm})}^{(n)}$ starting from the super Hopf algebra $P_{B}^{(n)}$.Comment: 9 pages. Contribution to the 6th Paanhellenic conference in Algebra and Number Theory, Aristotle University of Thessaloniki, Thessaloniki, 10-12 June 200

Ternary Poisson algebra for the non degenerate three dimensional Kepler Coulomb potential

In the three dimensional flat space any classical Hamiltonian, which has five functionally independent integrals of motion, including the Hamiltonian, is characterized as superintegrable. Kalnins, Kress and Miller have proved that, in the case of non degenerate potentials, i.e potentials depending linearly on four parameters, with quadratic symmetries, posses a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary parafermionic-like quadratic Poisson algebra with five generators. The Kepler Coulomb potential that was introduced by Verrier and Evans is a special case of superintegrable system, having two independent integrals of motion of fourth order among the remaining quadratic ones. The corresponding Poisson algebra of integrals is a quadratic one, having the same special form, characteristic to the non degenerate case of systems with quadratic integrals.Comment: 13 Pages, Contribution to the 4th Workshop on Group Analysis of Differential Equations and Integrable Systems, Protaras, Cyprus, Oct. 200

Generalized deformed oscillator for vortices in superfluid films

The algebra of observables of a system of two identical vortices in a superfluid thin film is described as a generalized deformed oscillator with a structure function containing a linear (harmonic oscillator) term and a quadratic term. In contrast to the deformed oscillators occuring in other physical systems (correlated fermion pairs in a single-$j$ nuclear shell, Morse oscillator), this oscillator is not amenable to perturbative treatment and cannot be approximated by quons. From the mathematical viewpoint, this oscillator provides a novel boson realization of the algebra su(1,1).Comment: 12 pages, LaTe

Quadratic algebras for three dimensional non degenerate superintegrable systems with quadratic integrals of motion

The three dimensional superintegrable systems with quadratic integrals of motion have five functionally independent integrals, one among them is the Hamiltonian. Kalnins, Kress and Miller have proved that in the case of non degenerate potentials there is a sixth quadratic integral, which is linearly independent of the other integrals. The existence of this sixth integral imply that the integrals of motion form a ternary {parafermionic-like} quadratic Poisson algebra with five generators. We show that in all the non degenerate cases (with one exception) there are at least two subalgebras of three integrals having a Poisson quadratic algebra structure, which is similar to the two dimensional case.Comment: 21 pages, Detailed version of the talk given at the XXVII Colloquium on Group Theoretical Methods in Physics, Yerevan, Armenia, Aug. 200

Bosonisation and Parastatistics: An Example and an Alternative Approach

Definitions of the parastatistics algebras and known results on their Lie (super)algebraic structure are reviewed. The notion of super-Hopf algebra is discussed. The bosonisation technique for switching a Hopf algebra in a braided category ${}_{H}\mathcal{M}$ ($H$: a quasitriangular Hopf algebra) into an ordinary Hopf algebra is presented and it is applied in the case of the parabosonic algebra. A bosonisation-like construction is also introduced for the same algebra and the differences are discussed.Comment: 13 pages, Contribution to "AGMF: Algebra, Geometry, and Mathematical Physics", Baltic-Nordic Workshop: Lund, 12-14 October, 200

Quantum Groups and Their Applications in Nuclear Physics

Quantum algebras are a mathematical tool which provides us with a class of symmetries wider than that of Lie algebras, which are contained in the former as a special case. After a self-contained introduction to the necessary mathematical tools (q-numbers, q-analysis, q-oscillators, q-algebras), the SUq(2) rotator model and its extensions, the construction of deformed exactly soluble models (u(3)>so(3) model, Interacting Boson Model, Moszkowski model), the 3-dimensional q-deformed harmonic oscillator and its relation to the nuclear shell model, the use of deformed bosons in the description of pairing correlations, and the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, are discussed in some detail. A brief description of similar applications to the structure of molecules and of atomic clusters, as well as an outlook are also given.Comment: 82 pages, LaTeX, review articl

Parafermionic and Generalized Parafermionic Algebras

The general properties of the ordinary and generalized parafermionic algebras are discussed. The generalized parafermionic algebras are proved to be polynomial algebras. The ordinary parafermionic algebras are shown to be connected to the Arik-Coon oscillator algebras.Comment: 12 pages, LaTeX. Presented at the International Conference on Mathematical Physics (Istanbul 1997). To appear in the Proceeding

Quasi-Exactly Soluble Potentials and Deformed Oscillators

It is proved that quasi-exactly soluble potentials corresponding to an oscillator with harmonic, quartic and sextic terms, for which the $n+1$ lowest levels of a given parity can be determined exactly, may be approximated by WKB equivalent potentials corresponding to deformed anharmonic oscillators of SU$_q$(1,1) symmetry, which have been used for the description of vibrational spectra of diatomic molecules. This connection allows for the immediate approximate determination of the levels of the same parity lying above the lowest $n+1$ known levels, as well as of all levels of the opposite parity. Such connections are not possible in the cases of the q-deformed oscillator, the Q-deformed oscillator, and the modified P\"oschl-Teller potential with SU(1,1) symmetry.Comment: 15 pages, LaTeX; to appear in the Proceedings of the 6th Hellenic Symposium on Nuclear Physics (Piraeus, Greece, 26-27 May 1995