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arXiv · 1810.00130

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

Julien Gaboriaud·Luc Vinet·Stéphane Vinet·Alexei Zhedanov
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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.

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Julien Gaboriaud, Luc Vinet, Stéphane Vinet, Alexei Zhedanov. 2018-09-29. The dual pair $Pin(2n)\times\mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra. https://doi.org/10.1016/j.nuclphysb.2018.10.011

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