The dual pair Pin(2n)×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 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.