Tridiagonal pairs of q-Racah type, the double lowering operator ψ , and the quantum algebra Uq(sl2)Uq(sl2)

Let KK denote an algebraically closed field and let V   denote a vector space over KK with finite positive dimension. We consider an ordered pair of linear transformations A:V→VA:V→V and A⁎:V→VA⁎:V→V that satisfy the following four conditions: (i) Each of A,A⁎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+1A⁎Vi⊆Vi−1+Vi+Vi+1 for 0⩽i⩽d0⩽i⩽d, where V−1=0V−1=0 and Vd+1=0Vd+1=0; (iii) there exists an ordering {Vi⁎}i=0δ of the eigenspaces of A⁎A⁎ such that AVi⁎⊆Vi−1⁎+Vi⁎+Vi+1⁎ for 0⩽i⩽δ0⩽i⩽δ, where V−1⁎=0 and Vδ+1⁎=0; (iv) there does not exist a subspace W of V   such that AW⊆WAW⊆W, A⁎W⊆WA⁎W⊆W, W≠0W≠0, W≠VW≠V. We call such a pair a tridiagonal pair on V  . It is known that d=δd=δ; to avoid trivialities assume d⩾1d⩾1. We assume that A,A⁎A,A⁎ belongs to a family of tridiagonal pairs said to have q  -Racah type. This is the most general type of tridiagonal pair. Let {Ui}i=0d and {Ui⇓}i=0d denote the first and second split decompositions of V  . In an earlier paper we introduced the double lowering operator ψ:V→Vψ:V→V. One feature of ψ   is that both ψUi⊆Ui−1ψUi⊆Ui−1 and ψUi⇓⊆Ui−1⇓ for 0⩽i⩽d0⩽i⩽d, where U−1=0U−1=0 and U−1⇓=0. Define linear transformations K:V→VK:V→V and B:V→VB:V→V such that (K−qd−2iI)Ui=0(K−qd−2iI)Ui=0 and (B−qd−2iI)Ui⇓=0 for 0⩽i⩽d0⩽i⩽d. Our results are summarized as follows. Using ψ, K, B   we obtain two actions of Uq(sl2)Uq(sl2) on V  . For each of these Uq(sl2)Uq(sl2)-module structures, the Chevalley generator e acts as a scalar multiple of ψ  . For each of the Uq(sl2)Uq(sl2)-module structures, we compute the action of the Casimir element on V. We show that these two actions agree. Using this fact, we express ψ   as a rational function of K±1,B±1K±1,B±1 in several ways. Eliminating ψ from these equations we find that K and B are related by a quadratic equation.