Finite-dimensional irreducible modules of the Bannai–Ito algebra at characteristic zero

Assume that F is algebraically closed with characteristic 0. A central extension BI of the Bannai–Ito algebras is a unital associative F-algebra generated by X, Y, Z, and the relations assert that each of {X,Y}-Z,{Y,Z}-X,{Z,X}-Yis central in BI. In this paper, we classify the finite-dimensional irreducible BI-modules up to isomorphism. As we will see, the elements X, Y, Z are not always diagonalizable on finite-dimensional irreducible BI-modules.