Bannai–Ito algebras and the universal R -matrix of $$\pmb {\mathfrak {osp}}(1|2)$$osp(1|2)

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 $${\mathfrak {S}}_n$$.