Parabosonic Algebra Research Articles

Families of 2D superintegrable anisotropic Dunkl oscillators and algebraic derivation of their spectrum

We generalize the construction of integrals of motion for quantum superintegrable models and the deformed oscillator algebra approach. This is presented in the context of 1D systems admitting ladder operators satisfying a parabosonic algebra involving reflection operators and more generally extended oscillator algebras with grading. We apply the construction on two-dimensional oscillators. We also introduce two new superintegrable Hamiltonians that are the anisotropic Dunkl and the singular Dunkl oscillators. Integrals are constructed by extending the approach of Daskaloyannis to include grading. An algebraic derivation of the energy spectra of the two models is presented, making use of finite dimensional unitary representations. We show how the spectra divide into sectors, and make comparisons with the physical case.

The Bannai-Ito polynomials as Racah coefficients of the $sl_{-1}(2)$ algebra

The Bannai-Ito polynomials are shown to arise as Racah coeffi- cients for sl−1(2). This Hopf algebra has four generators including an involu- tion and is defined with both commutation and anticommutation relations. It is also equivalent to the parabosonic oscillator algebra. The coproduct is used to show that the Bannai-Ito algebra acts as the hidden symmetry algebra of the Racah problem for sl−1(2). The Racah coefficients are recovered from a related Leonard pair.

Gradings, Braidings, Representations, Paraparticles: Some Open Problems

A research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings and braided group structures present in the various parastatistical algebraic models. The second part of the proposal aims at refining and utilizing a previously published methodology for the study of the Fock-like representations of the parabosonic algebra, in such a way that it can also be directly applied to the other parastatistics algebras. Finally, in the third part, a couple of Hamiltonians is proposed, suitable for modeling the radiation matter interaction via a parastatistical algebraic model.

From slq(2) to a Parabosonic Hopf Algebra

A Hopf algebra with four generators among which an involution (reflection) operator, is introduced. The defining relations involve commutators and anticommutators. The discrete series representations are developed. Designated by $sl_{-1}(2)$, this algebra encompasses the Lie superalgebra $osp(1|2)$. It is obtained as a $q=-1$ limit of the $sl_q(2)$ algebra and seen to be equivalent to the parabosonic oscillator algebra in irreducible representations. It possesses a noncocommutative coproduct. The Clebsch-Gordan coefficients (CGC) of $sl_{-1}(2)$ are obtained and expressed in terms of the dual -1 Hahn polynomials. A generating function for the CGC is derived using a Bargmann realization.

Variants of bosonization in parabosonic algebra: the Hopf and super-Hopf structures in parabosonic algebra

Parabosonic algebra in finite or infinite degrees of freedom is considered as a -graded associative algebra, and is shown to be a -graded (or super) Hopf algebra. The super-Hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the osp(1/2n) Lie superalgebraic structure. The notion of super-Hopf algebra is equivalently described as a Hopf algebra in the braided monoidal category . The bosonization technique for switching a Hopf algebra in the braided monoidal category (where H is a quasitriangular Hopf algebra) into an ordinary Hopf algebra is reviewed. In this paper, we prove that for the parabosonic algebra PB, beyond the application of the bosonization technique to the original super-Hopf algebra, a bosonization-like construction is also achieved using two operators, related to the parabosonic total number operator. Both techniques switch the same super-Hopf algebra PB to an ordinary Hopf algebra, thus producing two different variants of PB, with an ordinary Hopf structure.

Homogeneous Algebras, Statistics and Combinatorics

After some generalities on homogeneous algebras, we give a formula connecting the Poincare series of a homogeneous algebra with the homology of the corresponding Koszul complex generalizing thereby a standard result for quadratic algebras. We then investigate two particular types of cubic algebras: The first one, called the parafermionic (parabosonic) algebra, is the algebra generated by the creation operators of the universal fermionic (bosonic) parastatics with D degrees of freedom while the second is the plactic algebra, i.e., the algebra of the plactic monoid with entries in {1, 2,..., D}. In the case D=2, we describe the relations with the cubic Artin–Schelter algebras. It is pointed out that the natural action of GL(2) on the parafermionic algebra for D=2 extends as an action of the quantum group GLp,q(2) on the generic cubic Artin–Schelter regular algebra of type S1; p and q being related to the Artin–Schelter parameters. It is claimed that this has a counterpart for any integer D⩾2.

Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan–Schwinger map

The aim of this paper is to show that there is a Hopf structure of the parabosonic and parafermionic algebras and this Hopf structure can generate the well-known Hopf algebraic structure of the Lie algebras, through a realization of Lie algebras using the parabosonic (and parafermionic) extension of the Jordan–Schwinger map. The differences between the Hopf algebraic and the graded Hopf superalgebraic structure on the parabosonic algebra are discussed.

Parabosons, Virasoro-type algebras and their deformations

Two-parameter (p,q) deformation of a parabosonic algebra underlying the two-particle Calogero model is considered. The Fock-space representation, Bargmann-Fock representation, coherent states, spectrum-generating algebra and a hidden supersymmetry are discussed. The paraboson algebra induces a centreless Virasoro-type algebra with a doubling of the space of generators. A two-parameter deformation of this Virasoro-type algebra is studied.