The dual pair [formula omitted], the Dirac equation and the Bannai–Ito algebra

Julien Gaboriaud,Luc Vinet,Stéphane Vinet,Alexei Zhedanov

doi:10.1016/j.nuclphysb.2018.10.011

Julien Gaboriaud, Luc Vinet + Show 2 more

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https://doi.org/10.1016/j.nuclphysb.2018.10.011

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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(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)×osp(1|2) symmetry of the massless Dirac equation in R2n. Dimensional reduction to Rn will provide an alternative to the Dirac–Dunkl equation as a model with Bannai–Ito symmetry.

Highlights

We wish to shed light on the result of the previous two sections by casting in a Howe duality context the observation that the Bannai-Ito algebra arises as the commutant of both osp(1|2) in U (osp(1|2)⊗3) and o(2)⊕3 in the considered spinorial representations of U (o(6))
This paper has offered a novel presentation of the Bannai-Ito algebra B(n) as the commutant of o(2) ⊕ · · · ⊕ o(2) in the spinorial representation of o(2n) associated to the Clifford algebra Cl2n
It has indicated how this picture can be elegantly related to the definition of B(n) as the centralizer in U (osp(1|2)⊗n) of the coproduct embedding of the Lie superalgebra osp(1|2) in osp(1|2)⊗n in the framework of the Howe duality associated to P in(2n), osp(1|2)

Summary

Introduction

The Bannai-Ito algebra B(n) can be presented in terms of generators and relations [1, 2]. A key connection between the Lie superalgebra osp(1|2) and the Bannai-Ito algebra was made [3]. Like B(n), R(n) can be defined in a tensorial fashion as the algebra formed by the intermediate Casimir operators associated to the n-fold tensor product of the Lie algebra sl(2) with itself. An embedding of the higher rank R(n) Racah algebra in the B(n) Bannai-Ito algebra will be explicitly given, linking the construction presented here to the one in [16]. Let J0, J± be respectively the even and odd generators of the algebra, obeying the relations [J0, J±] = ±J±,.

The rank 1 Bannai-Ito algebra as a commutant

The Dirac model and Howe duality

Dimensional reduction

The higher rank Bannai-Ito algebra as a commutant

Conclusion