From $s{l}_q(2)$ to a Parabosonic Hopf Algebra

Abstract

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 $s{l}_{-1}(2)$, this algebra encompasses the Lie superalgebra $osp(1|2)$. It is obtained as a $q=-1$ limit of the $s{l}_q(2)$ 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 $s{l}_{-1}(2)$ are obtained and expressed in terms of the dual -1 Hahn polynomials. A generating function for the CGC is derived using a Bargmann realization.

References in corpus (2)

• A Bochner Theorem for Dunkl Polynomials
• Dunkl shift operators and Bannai-Ito polynomials

Cited by in corpus (18)

• The Dunkl oscillator in the plane II : representations of the symmetry algebra
• The Dunkl oscillator in three dimensions
• Bispectrality of the Complementary Bannai-Ito Polynomials
• Supersymmetric Quantum Mechanics with Reflections
• The Bannai-Ito algebra and a superintegrable system with reflections on the 2-sphere
• Embeddings of the Racah Algebra into the Bannai-Ito Algebra
• The algebra of dual -1 Hahn polynomials and the Clebsch-Gordan problem of sl_{-1}(2)
• The Hahn superalgebra and supersymmetric Dunkl oscillator models
• An Algebraic Model for the Multiple Meixner Polynomials of the First Kind
• An infinite family of superintegrable Hamiltonians with reflection in the plane
• Deformed su(1,1) Algebra as a Model for Quantum Oscillators
• The quantum superalgebra ${\mathfrak{osp}}_q(1|2)$ and a $q$-generalization of the Bannai-Ito polynomials
• Generating functions for the osp(1|2) Clebsch-Gordan coefficients
• Completing the solution for the $OSp(1|2)$ spin chain
• Superintegrability and the dual $-1$ Hahn algebra in superconformal quantum mechanics
• A Howe correspondence for the algebra of the $\mathfrak{osp}(1|2)$ Clebsch-Gordan coefficients
• Generating functions for the Bannai-Ito polynomials
• Convolution identities for Dunkl orthogonal polynomials from the $\mathfrak{osp}(1|2)$ Lie superalgebra