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

TL;DR

This paper explores the algebraic structures underlying the Dirac equation in higher dimensions, revealing connections between the Bannai-Ito algebra, Howe duality, and symmetries in spinorial representations, with implications for models like the Dirac-Dunkl equation.

Contribution

It establishes a novel link between the Bannai-Ito algebra, Howe duality, and the $Pin(2n)\times\mathfrak{osp}(1|2)$ symmetry in the context of the Dirac equation and its dimensional reductions.

Findings

Identification of the Bannai-Ito algebra as a centralizer and commutant in different algebraic frameworks.

Embedding of the Racah algebra within the commutant related to the Bannai-Ito algebra.

Dimensional reduction yields an alternative model with Bannai-Ito symmetry.

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.