Tridiagonal pairs of q-Racah type

AbstractLet F denote an algebraically closed field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A∗:V→V that satisfy the following conditions: (i) each of A,A∗ is diagonalizable; (ii) there exists an ordering {Vi}i=0d of the eigenspaces of A such that A∗Vi⊆Vi−1+Vi+Vi+1 for 0⩽i⩽d, where V−1=0 and Vd+1=0; (iii) there exists an ordering {Vi∗}i=0δ of the eigenspaces of A∗ such that AVi∗⊆Vi−1∗+Vi∗+Vi+1∗ for 0⩽i⩽δ, where V−1∗=0 and Vδ+1∗=0; (iv) there is no subspace W of V such that AW⊆W, A∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ. For 0⩽i⩽d let θi (resp. θi∗) denote the eigenvalue of A (resp. A∗) associated with Vi (resp. Vi∗). The pair A,A∗ is said to have q-Racah type whenever θi=a+bq2i−d+cqd−2i and θi∗=a∗+b∗q2i−d+c∗qd−2i for 0⩽i⩽d, where q,a,b,c,a∗,b∗,c∗ are scalars in F with q,b,c,b∗,c∗ nonzero and q2∉{1,−1}. This type is the most general one. We classify up to isomorphism the tridiagonal pairs over F that have q-Racah type. Our proof involves the representation theory of the quantum affine algebra Uq(slˆ2)