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Let \K denote an algebraically closed field and let V denote a vector spaceover \K with finite positive dimension. We consider an ordered pair of lineartransformations A:V\to V,A*:V \to V that satisfy the followingconditions:(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\subseteqV_{i-1}+V_i+V_{i+1} for 0\leq i\leq d, where V_{-1}=0 and V_{d+1}=0;(iii)thereexists an ordering {V*_i}_{i=0}^\delta of the eigenspaces of A* such that AV*_i\subseteq V*_{i-1}+V*_i+V*_{i+1} for 0\leq i\leq\delta, where V*_{-1}=0 andV*_{\delta+1}=0;(iv)there does not exist a subspace W of V such thatAW\subseteq W,A*W\subseteq W,W\neq 0,W\neq V. We call such a pair a tridiagonalpair on V. It is known that d=\delta; to avoid trivialities assume d\geq 1. Weassume that A,A* belongs to a family of tridiagonal pairs said to have q-Racahtype. This is the most general type of tridiagonal pair. Let {U_i}_{i=0}^d and{U_i^\Downarrow}_{i=0}^d denote the first and second split decompositions of V.In an earlier paper we introduced the double lowering operator \psi:V\to V. Onefeature of \psi is that both \psi U_i\subseteq U_{i-1} and \psiU_i^\Downarrow\subseteq U_{i-1}^\Downarrow for 0\leq i\leq d. Define lineartransformations K:V\to V and B:V\to V such that (K-q^{d-2i}I)U_i=0 and(B-q^{d-2i}I)U_i^\Downarrow=0 for 0\leq i\leq d. Our results are summarized asfollows. Using \psi,K,B we obtain two actions of Uq(sl2) on V. For each ofthese Uq(sl2)-module structures, the Chevalley generator e acts as a scalarmultiple of \psi. For each of the Uq(sl2)-module structures, we compute theaction of the Casimir element on V. We show that these two actions agree. Usingthis fact, we express \psi as a rational function of K^{\pm 1},B^{\pm 1} inseveral ways. Eliminating \psi from these equations we find that K,B arerelated by a quadratic equation.

Tridiagonal pairs of $q$-Racah type, the double lowering operator $\psi$, and the quantum algebra $U_q(\mathfrak{sl}_2)$