Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of Uq(sl2)

Let F denote an algebraically closed field; let q be a nonzero scalar in F such that q is not a root of unity; let d be a nonnegative integer; and let X, Y, Z be the equitable generators of Uq(sl2) over F . Let V denote a finite-dimensional irreducible Uq(sl2)-module with dimension d + 1, and let R denote the set of all linear maps from V to itself that act tridiagonally on the standard ordering of the eigenbases for each of X, Y, and Z. We show that R has dimension at most seven. Indeed, we show that the actions of 1, X, Y, Z, XY, YZ, and ZX on V give a basis for R when d ≥ 3.


Introduction
We characterize the linear operators that act tridiagonally with respect to appropriately ordered eigenbases for all three equitable generators of U q (sl 2 ) acting on its finite-dimensional irreducible modules. To state the main result, we first recall the equitable presentation of U q (sl 2 ). Throughout this paper, let F denote an algebraically closed field, and let q be a nonzero scalar in F such that q is not a root of unity.

Definition 1. [Definition 5.2] [1]
Let n X , n Y , n Z denote the following elements of U q (sl 2 ): Definition 2. Let V denote a vector space over F with dimension d + 1. By a decomposition of V, we mean a sequence {V i } d i=0 consisting of one-dimensional subspaces of V such that: Definition 4. Let 0 = q ∈ F , q 2 = 1; an LR pair A, B on V is said to be the q-Weyl type whenever: An LR triple A, B, C on V is said to be the q-Weyl type whenever the LR pairs A, B, B, C, and C, A all are the q-Weyl type.
Let A, B, C be an LR triple q-Weyl type on V. In [22], Nomura describes a family of linear maps that acts tridiagonally with respect to each of the (A, B), (B, C), and (C, A) decompositions for V.
The point of view of our work is quite different. To state the main result of this paper, we use the following definition.

Definition 5.
A square matrix is said to be tridiagonal whenever each nonzero entry lies on either the diagonal, the subdiagonal, or the superdiagonal. A square matrix is said to be lower bidiagonal whenever each nonzero entry lies on either the diagonal or the subdiagonal; a square matrix is said to be upper bidiagonal whenever each nonzero entry lies on either the diagonal or the superdiagonal.
Our main result is the following; an sl 2 analogue appears in [23].
Then, the following are equivalent.

(i)
Ψ acts on V as a linear combination of one, X, Y, Z, XY, YZ, and ZX.

(ii)
All three of the matrices representing Ψ with respect to standard X-, Y-, and Z-eigenbases are tridiagonal.

(iii)
Any two of the matrices representing Ψ with respect to standard X-, Y-, and Z-eigenbases are tridiagonal.
Moreover, one, X, Y, Z, XY, YZ, and ZX are linearly independent when dim V ≥ 3.

Standard Eigenbases for U q (sl 2 )-Modules
In this section, we recall the finite-dimensional U q (sl 2 )-modules and some distinguished bases.
The basis {u 0 , u 1 , . . . , u d } is called the standard X-eigenbasis of V d, .
Since the module V d,−1 can be treated similarly to V d,1 , we treat only the module V d,1 , and throughout this paper, we write V d to mean V d,1 . For any vector space V, End(V) is the F -algebra of all F -linear transformations from V to itself.
Proof. This is clear from Definition 6.

Lemma 4. [24]
With reference to Definition 1, let V d be a finite-dimensional irreducible U q (sl 2 )-module; the following hold: for V is said to be [τ] row whenever the following hold:

Lemma 5.
With reference to Lemma 2, the basis u = {u 0 , u 1 , . . . , u d } is the [X] row basis for V d .
Proof. Note that by Lemma 2, and: Moreover, note that from Lemma 3: For all integers k and for all nonnegative integers n, m, write: Definition 8. Let P and Q denote (d + 1) × (d + 1) matrices with entries P ij and Q ij , respectively, where: The matrix Q is a transition matrix from [X] row to [Z] row .

Lemma 6.
With reference to Lemma 2 and Definition 8, Proof. This is clear from Theorem 3.

Lemma 8.
With reference to Theorem 3, for all s ∈ U q (sl 2 ),

Proof. By Theorem 2 and elementary linear algebra, for
Similarly, we can prove the result for Y and Z. Since these formulas hold on generators, they must hold for all s ∈ U q (sl 2 ).

Linear Combinations of 1, X, Y, Z, Xy, Yz, Zx
In this section, we define the linear transformation A and describe the action of A on the bases u, v, and w given in the previous section, which we will use later to prove Theorem 1 and some special cases of this theorem.

Lemma 10. With reference to Lemma 2,
Proof. Performroutine calculations using the action of X, Y, and Z on the basis u in Lemma 2.
Lemma 11. With reference to Lemma 2 and Theorem 3, Proof. By elementary linear algebra and Lemma 7, Similarly, we can prove the other results. A = a I 1 + a x X + a y Y + a z Z + a xy XY + a yz YZ + a zx ZX.

Lemma 12.
With reference to Lemma 2 and Definition 9, the action of A on the basis u is given by: where: Proof. Use the actions of X, Y, and Z from Lemma 2 and the actions of XY, YZ, and ZX from Lemma 10 on the basis u to get the result.

Lemma 13.
With reference to Theorem 3 and Definition 9, the action of A on the basis v is given by: where: Proof. The actions of X, Y, and Z on the basis v are given in Theorem 3, and by Lemma 11, the actions of XY, YZ, and ZX on the basis v have the same coefficients of the actions of ZX, XY, and YZ on the basis u, respectively, which appear in Lemma 10. Now, expand the actions of these components on the basis v to get the result.

Lemma 14.
With reference to Theorem 3 and Definition 9, the action of A on the basis w is given by: where: Proof. The actions of X, Y, and Z on the basis w are given in Theorem 3, and by Lemma 11, the actions of XY, YZ, and ZX on the basis w have the same coefficients of the actions of YZ, ZX, and XY on the basis u, respectively, which appear in Lemma 10. Now, expand the actions of these components on the basis w to get the result. Proof. This is clear from Lemmas 12-14.
Note that Lemmas 12-14 give that (i) implies (ii) in Theorem 1. Routine calculations using Lemmas 12-14 and taking the advantage of the symmetry of the actions of X, Y, Z on the bases u, v, w give expressions for a I , a x , a y , a z , a xy , a yz , a zx , which appear in the following corollaries.

Tridiagonal Operators
In this section, we prove that (iii) implies (i) in Theorem 1.
and a I , a x , a y , a z , a xy , a yz , a zx are related by Lemmas 12 and 13.

(ii)
Ψ acts on V d as A = a I 1 + a x X + a y Y + a z Z + a xy XY + a yz YZ + a zx ZX.
For (ii), expand the actions of X, Y, and Z on the basis u to verify that the action of A agrees with the action of Ψ on the basis u. Now, (iii) follows from (ii) since the operators in the sum act tridiagonally on any [Z] row basis by Theorem 3.
We can use the same argument to prove the result in Lemma 15 when Ψ acts tridiagonally on the bases v and w or when Ψ acts tridiagonally on the bases w and u.

Main Results
In this section, we prove the main result of this paper, and then, we give some special cases. Theorem 4. Let V be a finite-dimensional U q (sl 2 )-module. Fix a linear map Ψ : V → V. Then, the following are equivalent.

(i)
Ψ acts on V as a linear combination of one, X, Y, Z, XY, YZ, and ZX.

(ii)
All three of the matrices representing Ψ with respect to standard X-, Y-, and Z-eigenbases are tridiagonal.

(iii)
Any two of the matrices representing Ψ with respect to standard X-, Y-, and Z-eigenbases are tridiagonal.
Moreover, one, X, Y, Z, XY, YZ, and ZX are linearly independent when dim V ≥ 3.