Leonard pairs were introduced by Terwilliger in [1], which gave some examples to illustrate how Leonard pairs arise in representation theory, combinatorics, and the theory of orthogonal polynomials. Because these polynomials frequently arise in connection with the finite-dimensional representations of good Lie algebras and quantum groups, it is natural to find Leonard pairs associated with these algebraic objects. Leonard pairs of Krawtchouk type have been described in [2] using split basis and normalized semisimple generators of sl₂. Leonard pairs of q-Krawtchouk type have been described in [3] using split basis of U_q(sl₂). Recently, Alnajjar and Curtin in [4] gave general construction of Leonard pairs of Racah, Hahn, dual Hahn and Krawtchouk type using equitable basis of sl₂. Alnajjar in [5, 6] gave general construction of Leonard pairs of q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk, and affine q-Krawtchouk type using equitable generators of U_q(sl₂). The Leonard pairs and Leonard triples of q-Racah type from the quantum algebra U_q(sl₂) were also discussed by Hou in [7] and [8]. Equitable presentations for U_q(sl₂) were introduced in [9].

Given a Leonard pair it is often more natural to work with a split basis rather than a standard basis. In this paper, we illustrate this with an example based on the quantum algebra ν_q(sl₂). Let M denote a finite-dimensional irreducible ν_q(sl₂)-module and assume $\mathbb{A}$ (resp. $\mathbb{B}$) is an arbitrary linear combination of F, K (resp. EH^-1, H^-1). We give the necessary and sufficient conditions on the coefficients for $\mathbb{A}$, $\mathbb{B}$ to act on M as a Leonard pair.

The rest of this paper is organized as follows. In Section 2, we introduce some facts concerning the Leonard pairs. In Section 3, we recall some facts concerning irreducible finite-dimensional ν_q(sl₂)-modules. In Section 4, we define two linear transformations $\mathbb{A}$ and $\mathbb{B}$ using the elements in ν_q(sl₂) and describe their properties. At last, we characterize when the pair $\mathbb{A}$, $\mathbb{B}$ is a Leonard pair.

In this section, we will recall the definitions and some related facts concerning Leonard pairs, for more details about the Leonard pairs can be found in [3]. Throughout this paper $\mathcal{F}$ denotes an algebraically closed field. Fix a nonzero scalar q∈ $\mathcal{F}$ which is not a root of unity. M_d+1($\mathcal{F}$) denote the $\mathcal{F}$-algebra consisting of all (d+1) by (d+1) matrices having rows and columns indexed by 0, 1, 2, ..., d for a nonnegative integer d.

Let V denote a $\mathcal{F}$-vector space of dimension d+1. Let End(V) denote the $\mathcal{F}$-algebra consisting of all linear transformations from V to V. Let {v_i}_i=0^d denote a basis for V. For $\mathbb{A}$ ∈ End(V) and X∈ M_d+1($\mathcal{F}$), we say X represents $\mathbb{A}$ with respect to {v_i}_i=0^d whenever $\mathbb{A}$${v_j} = \sum\limits_{i = 0}^d {{X_{ij}}{v_i}} $for 0≤j≤d, where X_ij is the element in the matrix X.

A square matrix is said to be tridiagonal if each nonzero entry lies on either the diagonal, the subdiagonal, or the superdiagonal. A tridiagonal matrix is said to be irreducible if each entry on the subdiagonal is nonzero and each entry on the superdiagonal is nonzero. We now define a Leonard pair.

Definition 2.1   Let V be a vector space over $\mathcal{F}$ with finite positive dimension. A Leonard pair on V is an ordered pair of linear transformations $\mathbb{A}$:V→ V and $\mathbb{A}$^*:V→ V that satisfy both the conditions below.

(1) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal.

(2) There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal.

There are so many examples of Leonard pairs which arise in representation theory, combinatorics, and the theory of orthogonal polynomials, for details can be found in [3].

Given a Leonard pair $\mathbb{A}$, $\mathbb{A}$^*, it is natural to represent one of $\mathbb{A}$, $\mathbb{A}$^* by an irreducible tridiagonal matrix and the other by a diagonal matrix. In order to distinguish the two representations, Terwilliger introduced the standard basis and split basis for this pair in [3]. A square matrix is said to be lower bidiagonal whenever every nonzero entry lies on either the diagonal or the subdiagonal. A lower bidiagonal is said to be irreducible lower bidiagonal whenever each entry on the subdiagonal is nonzero. A matrix is upper bidiagonal (resp. irreducible upper bidiagonal) whenever its transpose is lower bidiagonal (resp. irreducible lower bidiagonal).

Definition 2.2   Let V denote a vector space over $\mathcal{F}$ with finite positive dimension and let $\mathbb{A}$, $\mathbb{A}$^* denote a Leonard pair on V. A standard basis for this pair is a basis for V with respect to which the matrix representing $\mathbb{A}$ is irreducible tridiagonal and the matrix representing $\mathbb{A}$^* is diagonal.

Definition 2.3   Let V denote a vector space over $\mathcal{F}$ with finite positive dimension and let $\mathbb{A}$, $\mathbb{A}$^* denote a Leonard pair on V. A split basis for this pair is a basis for V with respect to which the matrix representing $\mathbb{A}$ is irreducible lower bidiagonal and the matrix representing $\mathbb{A}$^* is irreducible upper bidiagonal.

The following theorem in [3] provides a way to recognize a Leonard pair.

Theorem 2.4   (see [3]) Let V denote a vector space over $\mathcal{F}$ with finite positive dimension. Let $\mathbb{A}$ : V → V and $\mathbb{B}$ : V → V denote linear transformations. Let us assume there exists a basis for V with respect to which the matrices representing $\mathbb{A}$ and $\mathbb{B}$ have the following form.

$ A: \left( \begin{array}{ccccc} θ_0 & & & & \\ 1 & θ_1& & & \\ & 1 & θ_2 & & \\ & & \ddots & \ddots & \\ & & & 1 & θ_d \\ \end{array} \right)\hbox{, } B:\left( \begin{array}{ccccc} θ^*_0 &φ_1 & & & \\ & θ^*_1&φ_2 & & \\ & & θ^*_2 & \ddots & \\ & & & \ddots & φ_3 \\ & & & & θ^*_d \\ \end{array} \right) $

(2.1)

Then the pair $\mathbb{A}$, $\mathbb{B}$ is a Leonard pair on V if and only if there exist scalars ϕ_i (1≤i≤d) in $\mathcal{F}$ such that conditions (i)-(v) hold below.

(i) φ_i≠ 0, ϕ_i≠ 0, 1≤i≤d;

(ii) θ_i≠θ_j, θ_i^*≠θ_j^* if i≠ j, 0≤i, j≤d;

(iii) ${\varphi _i} = {\phi _1}\sum\limits_{h = 0}^{i - 1} {\frac{{{\theta _h} - {\theta _{d - h}}}}{{{\theta _0} - {\theta _d}}}} + \left( {\theta _i^ * - \theta _0^ * } \right)\left( {{\theta _{i - 1}} - {\theta _d}} \right), 1 \le i \le d;$

(iv) ${\phi _i} = {\varphi _1}\sum\limits_{h = 0}^{i - 1} {\frac{{{\theta _h} - {\theta _{d - h}}}}{{{\theta _0} - {\theta _d}}}} + \left( {\theta _i^ * - \theta _0^ * } \right)\left( {{\theta _{d - i + 1}} - {\theta _0}} \right), 1 \le i \le d;$

(v) The expressions $\frac{{{\theta _i}_{ - 2} - {\theta _{i + 1}}}}{{{\theta _i}_{ - 1} - {\theta _i}}}$, $\frac{{\theta _{i - 2}^ * - \theta _{i + 1}^ * }}{{\theta _{i - 1}^ * - \theta _i^ * }}$ are equal and independent of i for 2≤i≤d-1. In the rest of this paper, we will use the theorem to get Leonard pairs.

In this section, we recall some facts concerning irreducible finite-dimensional ν_q(sl₂)-modules in [10].

Definition 3.1   The quantum algebra ν_q(sl₂) is defined as the associative algebra (with 1 and over $\mathcal{F}$) with the generators E, F, K, K^-1, H, H^-1 and the following relations

$ \begin{eqnarray*} KK^{-1} = HH^{-1}= 1=K^{-1}K =H^{-1}H, KH =H K, \\ K E K^{-1} =q^2E, H E H^{-1} =q^2E , \\ K F K ^{-1}=q^{-2}F , H F H^{-1} =q^{-2}F, \\ EF-q^{-2}FE= q^{-2}(KH-1). \end{eqnarray*} $

Lemma 3.2   Let t≥ 1 be an integer. Then we have the following formulas in ν_q(sl₂),

$ \begin{align} & E{{F}^{t}}={{q}^{-2t}}{{F}^{t}}E+{{F}^{t-1}}[{{q}^{-4(t-1)}}{{(t)}_{q}}HK-{{(t)}_{{{q}^{-1}}}}]{{q}^{-2}}, \\ & F{{E}^{t}}={{q}^{2t}}{{E}^{t}}F+{{E}^{t-1}}[{{(t)}_{q}}-{{q}^{4(t-1)}}{{(t)}_{{{q}^{-1}}}}HK], \\ \end{align} $

where (t)_q=1+q²+...+q^2(t-1).

Lemma 3.3   Given an nonnegtative integer n and a, b∈ $\mathcal{F}$ with ab=q²ⁿ. Let M be a n+1-dimensional vector space with basis {m₀, m₁, ..., m_n}. We define the ν_q(sl₂)-action on M as follows,

$ K{m_i} = a{q^{ - 2i}}{m_i}H{m_i} = b{q^{ - 2i}}{m_i}\;\;{\rm{for}}\;\;0 \le i \le n, $

$ E{m_i} = \left\{ {\begin{array}{*{20}{c}} {\left[ {q{ - ^{ - 4\left( {i - 1} \right)}}{{\left( i \right)}_q}ab - {{\left( i \right)}_{{q^{ - 1}}}}} \right]{q^{ - 2}}{m_{i - 1}}, {\rm{if}}\;\;0 < i < n, }\\ {0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}\;\;i = 0, } \end{array}} \right. $

$ F{m_i} = \left\{ {\begin{array}{*{20}{c}} {{m_{i + 1}}, \;\;\;{\rm{if}}\;\;0 \le i < n, }\\ {0, \;\;\;\;\;\;\;\;{\rm{if}}\;\;i = n.} \end{array}} \right. $

Then M becomes a ν_q(sl₂)-module, we denote by M(n, a, b).

Theorem 3.4   Suppose that V is a finite dimensional irreducible ν_q(sl₂)-module with dimension n+1, then V is isomorphic to M(n, a, b) for some a, b∈ $\mathcal{F}$ with ab=q²ⁿ.

We will describe the construction of Leonard pairs from ν_q(sl₂)-modules by using generators of ν_q(sl₂) in the next section.

In this section, we define two linear transformations $\mathbb{A}$ and $\mathbb{B}$ of elements in ν_q(sl₂) and characterize when the pair $\mathbb{A}$, $\mathbb{B}$ is a Leonard pair.

Definition 4.1   Referring to Definition 3.1 and Lemma 3.3, let α, β denote nonzero scalars in $\mathcal{F}$. Then define two linear transformations $\mathbb{A}$, $\mathbb{B}$ as follows.

$ \begin{equation}\label{eq5} \mathbb{A}=α F+\frac{a^{-1}K}{q^2-1}, \mathbb{B}=\frac{b}{q^2-1}β E H^{-1}+\frac{bH^{-1}}{q^2-1}. \end{equation} $

(4.1)

Now we give the main result in this paper.

Theorem 4.2   Let n be an nonnegative integer and a, b∈ $\mathcal{F}$ with ab=q²ⁿ. Then the pair $\mathbb{A}$, $\mathbb{B}$ defined in (4.1) acts on M(n, a, b) as a Leonard pair provided αβ is not among q^-2, q^-4..., q^-2n.

To prove the above theorem, we apply Theorem 2.4. Before do this, we first give some lemmas.

Lemma 4.3   There exists a basis for M(n, a, b) with respect to which the matrices representing $\mathbb{A}$, $\mathbb{B}$ have the form of(2.1).

Proof   We can obtain this basis by modifying the basis {m₀, m₁, ..., m_n} given in Lemma 3.3. For 0≤i≤n, we define u_i=αⁱ m_i. We observe {u₀, u₁, ..., u_n} is a basis for M(n, a, b). The elements E, F, K, H act on this basis as follows.

$ \begin{array}{l} K{u_i} = a{q^{ - 2i}}{u_i}H{u_i} = b{q^{ - 2i}}{u_i}\;\;{\rm{for}}\;\;0 \le i \le n, \\ E{u_i} = \left\{ {\begin{array}{*{20}{c}} {\alpha \left[ {q{ - ^{ - 4\left( {i - 1} \right)}}{{\left( i \right)}_q}ab - {{\left( i \right)}_{{q^{ - 1}}}}} \right]{q^{ - 2}}{u_{i - 1}}, {\rm{if}}\;\;0 < i < n, }\\ {0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}\;\;i = 0, } \end{array}} \right.\\ F{u_i} = \left\{ {\begin{array}{*{20}{c}} {{\alpha ^{ - 1}}{u_{i + 1}}, \;\;\;{\rm{if}}\;\;0 \le i < n, }\\ {0, \;\;\;\;\;\;\;\;{\rm{if}}\;\;i = n.} \end{array}} \right. \end{array} $

(4.2)

Take ab=q²ⁿ into (4.2), we can get the coefficient of u_i-1 as below.

$ \begin{eqnarray*} % \nonumber to remove numbering (before each equation) α [q^{-4(i-1)}(i)_qab-(i)_{q^{-1}}]q^{-2}&=& α [q^{-4(i-1)}(i)_qq^{2n}-(i)_{q^{-1}}]q^{-2} \\ &=& α q^{-2i} [q^{2n-2i+2}(i)_q-(i)_{q^{-1}}q^{2i-2}]\\ &=& α q^{-2i} (i)_q (n-i+1)_{q}(q^{2}-1) \end{eqnarray*} $

Using these comments we can get

$ \mathbb{A}{{u}_{i}}={{u}_{i+1}}+\frac{{{q}^{-2i}}}{{{q}^{2}}-1}{{u}_{i}}, \ \ \ \mathbb{B}{{u}_{i}}=\alpha \beta {{(i)}_{q}}{{(n-i+1)}_{q}}{{u}_{i-1}}+\frac{{{a}^{-1}}{{q}^{2i}}}{{{q}^{2}}-1}{{u}_{i}} $

where u_-1=u_n+1=0. Thus, with respect to the basis {u₀, u₁, ..., u_n} the matrices representing $\mathbb{A}$, $\mathbb{B}$ are given in (2.1), where

$ \begin{eqnarray*} θ_i &=& \frac{q^{-2i}}{q^2-1}, \hbox{ }θ^*_i=\frac{q^{2i}}{q^2-1}\hbox{ }(0≤ i≤ n), \\ φ_i &=& αβ(i)_q(n-i+1)_q (0≤ i≤ n). \end{eqnarray*} $

Lemma 4.4   Referring to Lemma 4.3, the following two equations hold.

$ \begin{eqnarray*} % \nonumber to remove numbering (before each equation) \frac{θ_{i-2}-θ_{i+1}}{θ_{i-1}-θ_i}&=& q^2+q^{-2}+1, \\ \\ \frac{θ_{i-2}^*-θ_{i+1}^*}{θ_{i-1}^*-θ_i^*} &=& q^2+q^{-2}+1. \end{eqnarray*} $

Proof   Immediate from Lemma 4.3 and a simple calculation.

Lemma 4.5   Referring to Lemma 4.3, the scalars θ_i also satisfy the following equation.

$ \begin{eqnarray}\label{eq2} % \nonumber to remove numbering (before each equation) \sum\limits_{h=0}^{i-1}\frac{θ_h-θ_{n-h}}{θ_0-θ_n}&=&\frac{(i)_q(n-i+1)_q}{(n)_q}. \end{eqnarray} $

(4.3)

Proof   Using the sum of the geometric progression, we have

$ \begin{equation}\label{equ} (q^2-1)(t)_q=q^{2t}-1. \end{equation} $

(4.4)

Then from (4.4), the equation (4.3) holds.

Proof of Theorem 4.1   Define ${{\phi }_{i}}={{(i)}_{q}}{{(n-i+1)}_{q}}(\alpha \beta -{{q}^{-2(n-i+1)}})$, 1≤ i≤ n. Let us assume αβ is not among q^-2, q^-4..., q^-2n. Then the above scalars θ_i, θ_i^*, φ_i, ϕ_i satisfy conditions (i)-(v) of Theorem 2.4 by Lemmas 4.3, 4.4 and 4.5.

Remark 1   Applying Theorem 2.4 we find the pair $\mathbb{A}$, $\mathbb{B}$ acts on M(n, a, b) as a Leonard pair. With respect to the basis {u₀, u₁, ..., u_n}, the matrix representing $\mathbb{A}$ (resp. $\mathbb{B}$) is irreducible lower bidiagonal (resp. irreducible upper bidiagonal). Therefore this basis is a split basis for $\mathbb{A}$, $\mathbb{B}$ in view of Definition 2.3.

Remark 2   By the classification of Leonard pairs in [11], those with $\frac{{{\theta }_{i-2}}-{{\theta }_{i+1}}}{{{\theta }_{i-1}}-{{\theta }_{i}}}={{q}^{2}}+{{q}^{-2}}+1$ are the families q-Racah, q-Hahn, dual q-Hahn, quantum q-Krawtchouk, affine q- Krawtchouk, q-Krawtchouk, or dual q-Krawtchouk, and since the pair $\mathbb{A}$, $\mathbb{A}$ has this property(see Lemma 4.4), it's easy to show that this pair is of quantum q-Krawtchouk type.