The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra

Abstract

The Bannai-Ito algebra can be defined as the centralizer of the coproduct embedding of $\mathfrak{osp}(1|2)$ in $\mathfrak{osp}(1|2{)}^{\otimes n}$. It will be shown that it is also the commutant of a maximal Abelian subalgebra of $\mathfrak{o}(2n)$ 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 $Pin(2n)\times \mathfrak{osp}(1|2)$ symmetry of the massless Dirac equation in ${\mathbb{R}}^{2n}$. Dimensional reduction to ${\mathbb{R}}^n$ will provide an alternative to the Dirac-Dunkl equation as a model with Bannai-Ito symmetry.

arXiv comment16 pages

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Cited by in corpus (7)

• The Higgs and Hahn algebras from a Howe duality perspective
• Racah problems for the oscillator algebra, the Lie algebra ${\mathfrak{sl}}_n$, and multivariate Krawtchouk polynomials
• The $q$-Higgs and Askey-Wilson algebras
• The dual pair $(U_q(\mathfrak{su}(1,1)),{\mathfrak{o}}_{q^{1/2}}(2n))$, $q$-oscillators and Askey-Wilson algebras
• Howe duality and algebras of the Askey-Wilson type: an overview
• 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