Abstract
Let denote an algebraically closed field of characteristic zero and let
be some scalars in
. By the Racah algebra
associated with
, we mean the most general quadratic algebra with two algebraically independent generators
, which possesses presentations with ladder relations. In this paper, we classify the finite-dimensional irreducible
-modules up to isomorphism by using the theory of the Leonard pairs. For a given irreducible
-module V with dimension d + 1, we give its corresponding isomorphism classes of Leonard pairs on V that have Racah type.
2010 Mathematics Subject Classification:
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).