author={Satoshi Tsujimoto and Luc Vinet and Alexei S. Zhedanov},
  journal={Symmetry Integrability and Geometry-methods and Applications},
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 en- compasses the Lie superalgebra osp(1j2). It is obtained as a q = 1 limit of the slq(2) algebra and seen to be equivalent to the parabosonic oscillator algebra in irreducible repre- sentations. It possesses a noncocommutative coproduct. The… 
The Bannai-Ito polynomials as Racah coefficients of the sl_{-1}(2) algebra
The Bannai-Ito polynomials are shown to arise as Racah coefficients for sl_{-1}(2). This Hopf algebra has four generators including an involution and is defined with both commutation and
The Bannai–Ito algebra and a superintegrable system with reflections on the two-sphere
A quantum superintegrable model with reflections on the two-sphere is introduced. Its two algebraically independent constants of motion generate a central extension of the Bannai–Ito algebra. The
The algebra of dual −1 Hahn polynomials and the Clebsch-Gordan problem of sl−1(2)
The algebra H of the dual −1 Hahn polynomials is derived and shown to arise in the Clebsch-Gordan problem of sl−1(2). The dual −1 Hahn polynomials are the bispectral polynomials of a discrete
Deformed su(1;1) Algebra as a Model for Quantum Oscillators
The Lie algebra su(1; 1) can be deformed by a reflection operator, in such a way that the positive discrete series representations of su(1; 1) can be extended to representa- tions of this deformed
A Laplace-Dunkl Equation on S2 and the Bannai–Ito Algebra
The analysis of the $${\mathbb{Z}_2^{3}}$$Z23 Laplace-Dunkl equation on the 2-sphere is cast in the framework of the Racah problem for the Hopf algebra sl−1(2). The related Dunkl-Laplace operator is
The Hahn superalgebra and supersymmetric Dunkl oscillator models
A supersymmetric extension of the Hahn algebra is introduced. This quadratic superalgebra, which we call the Hahn superalgebra, is constructed using the realization provided by the Dunkl oscillator
Embeddings of the Racah algebra into the Bannai-Ito algebra
Embeddings of the Racah algebra into the Bannai-Ito algebra are proposed in two realiza- tions. First, quadratic combinations of the Bannai-Ito algebra generators in their standard realization on the
The Dunkl Oscillator in the Plane II: Representations of the Symmetry Algebra
The superintegrability, wavefunctions and overlap coefficients of the Dunkl oscillator model in the plane were considered in the first part. Here finite-dimensional representations of the symmetry
An infinite family of superintegrable Hamiltonians with reflection in the plane
We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schrodinger equations admit the separation of
The Dunkl oscillator in the plane I : superintegrability, separated wavefunctions and overlap coefficients
The isotropic Dunkl oscillator model in the plane is investigated. The model is defined by a Hamiltonian constructed from the combination of two independent parabosonic oscillators. The system is


Graded structure and Hopf structures in parabosonic algebra. An alternative approach to bosonisation
Parabosonic algebra in infinite degrees of freedom is presen ted as a generalization of the bosonic algebra, from the viewpoints of both physics and mathematics. The notion of super-Hopf algebra is
It is shown that the deformed Heisenberg algebra involving the reflection operator R (R-deformed Heisenberg algebra) has finite-dimensional representations which are equivalent to representations of
Deformed Heisenberg algebra with reflection
A universality of the deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to
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
Quantum Algebras and q-Special Functions
Abstract A quantum-algebraic framework for many q-special functions is provided. The two-dimensional Euclidean quantum algebra, slq(2) and the q-oscillator algebra are considered. Realizations of
The oscillator-type realization is proposed for the continuous set of infinite-dimensional algebras of quantum operators on the two-dimensional sphere and hyperboloid. This realization is typical for
The su ( 2 ) α Hahn oscillator and a discrete Hahn-Fourier transform
We define the quadratic algebra su(2)α which is a one-parameter deformation of the Lie algebra su(2) extended by a parity operator. The odd-dimensional representations of su(2) (with representation
Representations and properties of para‐Bose oscillator operators. I. Energy position and momentum eigenstates
Para‐Bose commutation relations are related to the SL(2,R) Lie algebra. The irreducible representation Dα of the para‐Bose system is obtained as the direct sum Dβ⊕Dβ+1/2 of the representations of the
Finite oscillator models: the Hahn oscillator
A new model for the finite one-dimensional harmonic oscillator is proposed based upon the algebra . This algebra is a deformation of the Lie algebra extended by a parity operator, with the
The Clebsch-Gordan coefficients for the quantum group SμU(2) and q-Hahn polynomials
The tensor product of two unitary irreducible representations of the quantum group SμU(2) is decomposed in a direct sum of unitary irreducible representations with explicit realizations. The