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.

16 pages