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

Abstract

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 orderingWe 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 } d i=0 and {U ⇓ i } d i=0 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 ⇓ −1 = 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.

The compact presentation for the alternating central extension of the $q$-Onsager algebra

“…Investigate the Jordan canonical form for these actions. How does the Bockting double lowering operator [12,13] act on the Jordan blocks?…”

“…In [9], two automorphisms of O q are introduced that resemble the Lusztig automorphisms of U q ( sl 2 ); see also [27,30]. These automorphisms are used in [9] to obtain a PBW basis for O q , and they are used in [33] to describe the Bockting double lowering operator [12,13] of a tridiagonal pair. A Drinfeld type presentation of O q is obtained in [21], and this is used in [22] to realize O q as an ιHall algebra of the projective line.…”

Tridiagonal pairs of $q$-Serre type and their linear perturbations

“…We consider a tridiagonal pair (A, A * ) of q-Serre type; for such a pair the A and A * satisfy the q-Serre relations as shown in (3) and (4) below. There is a linear map K in the literature (see [7,Section 1.1] and [2,3,9]) that is used to describe how A and A * are related. We investigate a pair of linear maps (B, B * ) of the form…”