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

@article{Gaboriaud2018TheDP, title={The dual pair Pin(2n)×osp(1|2), the Dirac equation and the Bannai–Ito algebra}, author={Julien Gaboriaud and Luc Vinet and St'ephane Vinet and Alexei S. Zhedanov}, journal={Nuclear Physics B}, year={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 Abelian subalgebra of o ( 2 n ) in a spinorial representation and an embedding of the Racah algebra in this commutant will emerge. The connection between the two pictures for the Bannai–Ito algebra will be traced to the Howe duality which is embodied in the P i n ( 2 n ) × osp ( 1 | 2 ) symmetry of the… Expand

#### 8 Citations

The Higgs and Hahn algebras from a Howe duality perspective

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Abstract The Hahn algebra encodes the bispectral properties of the eponymous orthogonal polynomials. In the discrete case, it is isomorphic to the polynomial algebra identified by Higgs as the… Expand

The q-Higgs and Askey–Wilson algebras

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Abstract A q-analogue of the Higgs algebra, which describes the symmetry properties of the harmonic oscillator on the 2-sphere, is obtained as the commutant of the o q 1 / 2 ( 2 ) ⊕ o q 1 / 2 ( 2 )… Expand

The dual pair (Uq(su(1,1)),oq1/2(2n)), q-oscillators, and Askey-Wilson algebras

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The universal Askey–Wilson algebra AW(3) can be obtained as the commutant of Uq(su(1,1)) in Uq(su(1,1))⊗3. We analyze the commutant of oq1/2(2)⊕oq1/2(2)⊕oq1/2(2) in q-oscillator representations of… Expand

Howe Duality and Algebras of the Askey–Wilson Type: An Overview

- Mathematics, Physics
- 2019

The Askey-Wilson algebra and its relatives such as the Racah and Bannai-Ito algebras were initially introduced in connection with the eponym orthogonal polynomials. They have since proved ubiquitous.… Expand

The dual pair $\big(U_q(\mathfrak{su}(1,1)),\mathfrak{o}_{q^{1/2}}(2n)\big)$, $q$-oscillators and Askey-Wilson algebras

- Physics, Mathematics
- 2019

The universal Askey-Wilson algebra $AW(3)$ can be obtained as the commutant of $U_q(\mathfrak{su}(1,1))$ in $U_q(\mathfrak{su}(1,1))^{\otimes3}$. We analyze the commutant of… Expand

A Howe correspondence for the algebra of the osp(1|2) Clebsch-Gordan coefficients

- Physics, Mathematics
- 2020

Abstract Two descriptions of the dual −1 Hahn algebra are presented and shown to be related under Howe duality. The dual pair involved is formed by the Lie algebra o ( 4 ) and the Lie superalgebra… Expand

Superintegrability and the dual −1 Hahn algebra in superconformal quantum mechanics

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A two-dimensional superintegrable system of singular oscillators with internal degrees of freedom is identified and exactly solved. Its symmetry algebra is seen to be the dual $-1$ Hahn algebra which… Expand

Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$, and Multivariate Krawtchouk Polynomials

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The oscillator Racah algebra $\mathcal{R}_n(\mathfrak{h})$ is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra $\mathfrak{h}$. An… Expand

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