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

@article{Genest2012TheBP, title={The Bannai-Ito polynomials as Racah coefficients of the sl\_\{-1\}(2) algebra}, author={Vincent X. Genest and Luc Vinet and Alexei S. Zhedanov}, journal={arXiv: Mathematical Physics}, year={2012} }

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 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.

## 49 Citations

A Laplace-Dunkl Equation on S2 and the Bannai–Ito Algebra

- Mathematics
- 2015

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…

Embeddings of the Racah algebra into the Bannai-Ito algebra

- Physics, Mathematics
- 2015

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 equitable Racah algebra from three $\mathfrak {su}(1,1)$ algebras

- Mathematics, Physics
- 2013

The Racah algebra, a quadratic algebra with two independent generators, is central in the analysis of superintegrable models and encodes the properties of the Racah polynomials. It is the algebraic…

THE QUANTUM SUPERALGEBRA ospq(1|2) AND A q-GENERALIZATION OF THE BANNAI-ITO POLYNOMIALS

- Mathematics
- 2015

The Racah problem for the quantum superalgebra ospq(1|2) is considered. The intermediate Casimir operators are shown to realize a q- deformation of the Bannai-Ito algebra. The Racah coefficients of…

The non-symmetric Wilson polynomials are the Bannai-Ito polynomials

- Mathematics, Physics
- 2015

The one-variable non-symmetric Wilson polynomials are shown to coincide with the Bannai-Ito polynomials. The isomorphism between the corresponding degenerate double affine Hecke algebra of type…

Bannai–Ito algebras and the osp(1;2) superalgebra

- Mathematics
- 2016

The Bannai–Ito algebra B(n) of rank (n – 2) is defined as the algebra generated by the Casimir operators arising in the n-fold tensor product of the osp(1,2) superalgebra. The structure relations are…

Centralizers of the superalgebra $\mathfrak{osp}(1|2)$ : the Brauer algebra as a quotient of the Bannai–Ito algebra

- MathematicsJournal of Physics A: Mathematical and Theoretical
- 2019

We provide an explicit isomorphism between a quotient of the Bannai--Ito algebra and the Brauer algebra. We clarify also the connection with the action of the Lie superalgebra osp(1|2) on the…

The dual pair Pin(2n)×osp(1|2), the Dirac equation and the Bannai–Ito algebra

- Physics, MathematicsNuclear Physics B
- 2018

Abstract The Bannai–Ito algebra can be defined as the centralizer of the coproduct embedding of osp ( 1 | 2 ) in osp ( 1 | 2 ) ⊗ n . It will be shown that it is also the commutant of a maximal…

Bispectrality of the Complementary Bannai-Ito Polynomials

- Mathematics, Physics
- 2013

A one-parameter family of operators that have the complementary Bannai{ Ito (CBI) polynomials as eigenfunctions is obtained. The CBI polynomials are the kernel partners of the Bannai{Ito polynomials…

The Z(2)(n) Dirac-Dunkl operator and a higher rank Bannai-Ito algebra

- Mathematics
- 2016

Abstract The kernel of the Z 2 n Dirac–Dunkl operator is examined. The symmetry algebra A n of the associated Dirac–Dunkl equation on S n − 1 is determined and is seen to correspond to a higher rank…

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