Tridiagonal pairs of q-Racah type, the double lowering operator ψ, and the quantum algebra U_q(sl_2)

摘要

Let K denote an algebraically closed field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A: V → V and A*: V → V that satisfy the following four 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_(-i) = 0 and V_(d+1) = 0; (iii) there exists an ordering {V_i~* }_(i=0)~δ of the eigenspaces of A* such that AV_i~* ? V_(i-1)~* + V_i~* + V_(i+1)~* for 0 ≤i ≤ δ, where V~(-1)~*= 0 and V_(δ+1)~*=0; (iv) there does not exist a 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; to avoid trivialities assume d ≥ 1. We assume that 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 {U_i}_(i=0)~d and {U_i~↓}_(i=0)~d denote the first and second split decompositions of V. In an earlier paper we introduced the double lowering operator ψ: V → V. One feature of ψ is that both ψU_i ? U_(i-1) and ψU_i~↓? U_(i-1)~↓ for 0 ≤i ≤ d, where U_(-1) = 0 and U_(-i)~↓= 0. Define linear transformations K: V → V and B: V → V such that (K - q~(d-2i)I)U_i = 0 and (B - q~(d-2i)I)U_i~↓ = 0 for 0 ≤i≤ d. Our results are summarized as follows. Using ψ, K, B we obtain two actions of U_q(sl_2) on V. For each of these U_q(sl_2)-module structures, the Chevalley generator e acts as a scalar multiple of ψ. For each of the U_q (sl_2)-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~(±1) in several ways. Eliminating ψ from these equations we find that K and B are related by a quadratic equation.