Tridiagonal pairs of q-Racah type

Let 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 A Vi* ⊆ 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 A W ⊆ 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 + b q2 i - d + c qd - 2 i and θi* = a* + b* q2 i - d + c* qd - 2 i 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 (over(sl, ̂)2).