Tridiagonal pairs of q-Racah type

Abstract 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 { V i } i = 0 d of the eigenspaces of A such that A ∗ V i ⊆ V i − 1 + V i + V i + 1 for 0 ⩽ i ⩽ d , where V − 1 = 0 and V d + 1 = 0 ; (iii) there exists an ordering { V i ∗ } i = 0 δ of the eigenspaces of A ∗ such that A V i ∗ ⊆ V i − 1 ∗ + V i ∗ + V i + 1 ∗ for 0 ⩽ i ⩽ δ , where V − 1 ∗ = 0 and V δ + 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 θ i (resp. θ i ∗ ) denote the eigenvalue of A (resp. A ∗ ) associated with V i (resp. V i ∗ ). The pair A , A ∗ is said to have q-Racah type whenever θ i = a + b q 2 i − d + c q d − 2 i and θ i ∗ = a ∗ + b ∗ q 2 i − d + c ∗ q d − 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 q 2 ∉ { 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 U q ( sl ˆ 2 ) .