Abstract
Let 𝕂 denote an algebraically closed field of characteristic zero, and let α, β, γ be some scalars in 𝕂. By the Bannai/Ito algebra denoted 𝒜(α, β, γ), we mean the associative algebra over 𝕂 with generators x, y, z and relations xy + yx = z + α, yz + zy = x + β, zx + xz = y + γ. In this article, we classify the finite-dimensional irreducible 𝒜(α, β, γ)-modules up to isomorphism by using the theories of the Leonard pairs and the Leonard triples.
ACKNOWLEDGMENT
The authors would like to thank the referee who gave many valuable suggestions. The authors are grateful to professor P. Terwilliger and professor T. Ito for the advice they offered during their study of the q-tetrahedron algebra.