Bannai-Ito algebras and the universal R-matrix of osp(1|2)

TL;DR

This paper explores the structure of Bannai-Ito algebras as centralizers of osp(1|2) actions, constructing generators via the universal R-matrix and deriving their relations through coproducts and symmetry properties.

Contribution

It introduces a novel construction of Bannai-Ito algebra generators using the universal R-matrix of osp(1|2) and elucidates their algebraic relations.

Findings

Generators constructed from the universal R-matrix.

Derived structure relations from those of BI(3).

Connected algebraic structures with osp(1|2) embeddings.

Abstract

The Bannai-Ito algebra $BI(n)$ is viewed as the centralizer of the action of $\mathfrak{osp}(1|2)$ in the $n$-fold tensor product of the universal algebra of this Lie superalgebra. The generators of this centralizer are constructed with the help of the universal $R$-matrix of $\mathfrak{osp}(1|2)$. The specific structure of the $\mathfrak{osp}(1|2)$ embeddings to which the centralizing elements are attached as Casimir elements is explained. With the generators defined, the structure relations of $BI(n)$ are derived from those of $BI(3)$ by repeated action of the coproduct and using properties of the $R$-matrix and of the generators of the symmetric group $\mathfrak S_n$.