The Classification of Finite-Dimensional Irreducible Modules of Bannai/Ito Algebra

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.

Key Words:

• Anti-commutator spin algebra
• Bannai/Ito algebra
• Irreducible module
• Leonard triple

2010 Mathematics Subject Classification:

• 15A04
• 16G99
• 20G05
• 33C80
• 33C45

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.