The Classification Of Finite-dimensional Irreducible Modules Of Racah Algebra

Let K denote an algebraically closed field of characteristic zero and let d0,e1,e2 be some scalars in K.By the Racah algebra associated with d0,e1,e2 denoted by A(d0,e1,e2),we mean the most general quadratic algebra with two algebraically independent genera-tors x,y and relations x2y-2xyx + yx2 +(xy + yx)+ x2 + d0x + e2 =0,y2x-2yxy + xy2 +(yx + xy)+y2 + d0y + e1=0.In this paper we classify the finite-dimensional irreducible A(do,e1,e2)-modules up to isomorphism by using the theories of the Leonard pairs.The main results are as the following:1.We first show that the actions of x,y on V are diagonalizable,and form a Leonard pair.Then we obtain the eigenvalues of the actions of x,y on V,respectively and classify up to isomorphism the irreducible modules of Racah algebra.2.Let d ? 3 be an integer.For a given irreducible A(d0,e1,e2)-module V with dimension d+1,we give its corresponding isomorphism classes of Leonard pairs on V that have Racah type.