The classification of finite-dimensional irreducible modules of the Racah algebra

Abstract

Let $\mathbb{K}$ denote an algebraically closed field of characteristic zero and let $d_0,e_1,e_2$ be some scalars in $\mathbb{K}$. By the Racah algebra $\mathcal{A}$ associated with $d_0,e_1,e_2$, we mean the most general quadratic algebra with two algebraically independent generators $x,y$, which possesses presentations with ladder relations. In this paper, we classify the finite-dimensional irreducible $\mathcal{A}$-modules up to isomorphism by using the theory of the Leonard pairs. For a given irreducible $\mathcal{A}$-module V with dimension d + 1, we give its corresponding isomorphism classes of Leonard pairs on V that have Racah type.

Keywords:

• Racah algebra
• Leonard pair
• irreducible $\mathcal{A}$-module

2010 Mathematics Subject Classification:

• Primary 17C55
• Secondary 17B10
• 05E10

Acknowledgements

The authors are especially grateful to an anonymous referee, whose extensive and detailed comments have greatly improved the accuracy and completeness of this article. The authors are also grateful to Professor P. Terwilliger and Professor T. Ito for the advice they offered during their study of the q-tetrahedron algebra. This work was supported by the NSFC (No. 11471097) and the NSF of Hebei Province (No. A2017403010).