# A Linear Map Acts as a Leonard Pair with Each of the Generators ofUsl2

Authors: Hasan Alnajjar
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

Letℱ denote an algebraically closed field with a characteristic not two. Fix an integer d≥3; let x, y, and z be the equitable basis of sl2 over ℱ. Let V denote an irreducible sl2-module with dimension d+1; let A∈EndV. In this paper, we show that if each of the pairs A,x, A,y, and A,z acts on V as a Leonard pair, then these pairs are of Krawtchouk type. Moreover, A is a linear combination of 1, x, y, and z.


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## Body

## 1. Introduction

In this section, we recall some facts concerning Leonard pairs. Leonard pairs were introduced by P. Terwilliger [1] to extend the algebraic approach of Bannai and Ito [2] to a result of D. Leonard concerning the sequences of orthogonal polynomials with finite support, for which the dual sequence of polynomials is also a sequence of orthogonal polynomials [3, 4]. These polynomials arise in connection with the finite-dimensional representations of certain Lie algebras and quantum groups, so one expects Leonard pairs to arise as well. Leonard pairs of Krawtchouk type have been constructed from finite-dimensional irreducible sl2-modules [5, 6]. Fix an integer d≥1. Throughout this paper ℱ shall denote an algebraically closed field with characteristic not two. Also, V shall denote an ℱ-vector space of dimension d+1, and Matd+1ℱ shall denote the ℱ-algebra of d+1×d+1 matrices with entries in ℱ having rows and columns indexed by 1, 2, …, d+1. A square matrix is said to be tridiagonal whenever every nonzero entry appears on, immediately above, or immediately below the main diagonal. A tridiagonal matrix is said to be irreducible whenever all entries immediately above and below the main diagonal are nonzero. A square matrix is said to be upper (resp. lower) bidiagonal whenever every nonzero entry appears on or immediately above (resp. below) the main diagonal.Definition 1.
LetV denote a vector space over ℱ with finite positive dimension. By a Leonard pair on V, we mean an ordered pair A, A∗, where A:V⟶V and A∗:V⟶V are linear transformations that satisfy both (i) and (ii) below:(i)
There exists a basis forV with respect to which the matrix representing A∗ is diagonal and the matrix representing A is irreducible tridiagonal(ii)
There exists a basis forV with respect to which the matrix representing A is diagonal and the matrix representing A∗ is irreducible tridiagonal
We remark that ifA, A∗ is a Leonard pair on V, then A∗, A is a Leonard pair on V, and for any scalars α, α∗, the pair A+αI, A∗+α∗I is also a Leonard pair on V.
For more details about Leonard pairs, see [7–14].Definition 2.
LetV denote a vector space over ℱ with dimension d+1. let vii=0d denote a basis for V, which satisfies condition (ii) of Definition 1. For 0≤i≤d, the vector vi is an eigenvector of A; let θi denote the corresponding eigenvalues. Let vi∗i=0d denote a basis for V, which satisfies condition (i) of Definition 1. For 0≤i≤d, the vector vi∗ is an eigenvector of A∗; let θi∗ denote the corresponding eigenvalues. Let the sequence aii=0d denote the diagonal of the matrix, which represents A with respect to vi∗i=0d. Let the sequence ai∗i=0d denote the diagonal of the matrix, which represents A∗ with respect to vii=0d.
The ordering ofθii=0d in Definition 2 is said to be standard. For a standard ordering θii=0d of eigenvalues of, the ordering θd−ii=0d is also standard and no further ordering is standard. A similar result applies for θi∗i=0d.Theorem 1 (see [15]).
LetA, A∗ be a Leonard pair on V; let θii=0d (resp. θi∗i=0d) be standard ordering of the eigenvalues of A (resp. A∗). Then, there exists a basis of V such that the matrices representing A, A∗ with respect to this basis are, respectively,(1)A=θ01θ11θ2⋱⋱1θd,A∗=θ0∗φ1θ1∗φ2θ2∗⋱⋱φdθd∗,for some sequence of scalars φ1, φ2, …, φd in ℱ, which we refer to as the first split sequence of A, A∗.Definition 3 (see [16]).
Letd denote a nonnegative integer. By a parameter array over ℱ of diameter d, we mean a sequence of scalars (θii=0d, θi∗i=0d; φjj=1d, ϕjj=1d) taken from ℱ that satisfy the following conditions:(2)θi≠θj,0≤i<j≤d,(3)θi∗≠θj∗,0≤i<j≤d,(4)φi≠0,1≤i≤d,(5)ϕi≠0,1≤i≤d,(6)φi=ϕ1∑h=0i−1θh−θd−hθ0−θd+θi∗−θ0∗θi−1−θd,1≤i≤d,(7)ϕi=φ1∑h=0i−1θh−θd−hθ0−θd+θi∗−θ0∗θd−i+1−θ0,1≤i≤d,(8)θi−2−θi+1θi−1−θi=θj−2∗−θj+1∗θj−1∗−θj∗,2≤i,j≤d−1.Theorem 2 (see [15]).
Letd denote a nonnegative integer; let B and B∗ denote matrices in Matd+1ℱ. Assume B is lower bidiagonal and B∗ is upper bidiagonal. Then, the following are equivalent:(i)
The pairB, B∗ is a Leonard pair in Matd+1ℱ.(ii)
There exists a parameter array (θii=0d, θi∗i=0d; φjj=1d, ϕjj=1d) over ℱ such that(9)Bi,i=θi,B∗i,i=θi∗,0≤i≤d,Bj,j−1B∗j−1,j=φj,1≤j≤d.
Suppose (i) and (ii) hold. Then, the parameter array in (ii) is uniquely determined byB, B∗.Theorem 3 (see [15]).
With reference to Definition2, let A, A∗ be a Leonard pair on V; let θii=0d (resp. θi∗i=0d) be standard ordering of the eigenvalues of A (resp. A∗). Via a standard basis, the matrix representing A and A∗ are(10)a0b00c1a1b10c2a2b2⋱⋱⋱⋱0cd−1ad−1bd−100cdadand diagθ0∗,θ1∗,…,θd∗, where(11)ai=θi+φiθi∗−θi−1∗+φi+1θi∗−θi+1∗,0≤i≤d,ai∗=θi∗+φiθi−θi−1+φi+1θi−θi+1,0≤i≤d,bi=φi+1Πh=0i−1θi∗−θh∗Πh=0iθi+1∗−θh∗,0≤i≤d−1,ci=ϕiΠh=0d−i−1θi∗−θd−h∗Πh=0d−iθi−1∗−θd−h∗,1≤i≤d,where φ0=0 and φd+1=0.Theorem 4 (see [17]).
LetV denote a vector space over ℱ with finite positive dimension. Let A, A∗ denote a Leonard pair on V. Then, there exists a sequence of scalars β, γ, γ∗, ϱ, ρ∗, ω, η, and η∗ taken from ℱ such that(12)A2A∗−βAA∗A+A∗A2−γAA∗+A∗A−ϱA∗=γ∗A2+ωA+ηI,(13)A∗2A−βA∗AA∗+AA∗2−γ∗A∗A+AA∗−ϱ∗A=γA∗2+ωA∗+η∗I.
The sequence is uniquely determined by the pairA, A∗ provided the dimension of V is at least 4.
Relations (12) and (13) are called the Askey–Wilson relations.


## 2. Leonard Pairs of Classical Type

In [18], the parameters arrays are classified into 13 families, each named for certain associated sequences of orthogonal polynomials. The four families, which arise in this paper, share certain property. Given a parameter array, let β be the common value of (8) minus one.Definition 4.
A parameter array is of classical type wheneverβ=2.Theorem 5 (see [18]).
A parameter array is of classical type if and only if it is of Racah, Hahn, dual Hahn, or Krawtchouk type.Theorem 6 (see [18], Example 5.10).
Fix nonzeroh, h∗∈ℱ and s, s∗, r1, r2, θ0, θ0∗∈ℱ such that r1+r2=s+s∗+d+1 and none of r1, r2, s∗−r1, s∗−r2 is equal to −i for 1≤i≤d and that neither of s, s∗ is equal to −i for 2≤i≤2d. Let(14)θi=θ0+hii+1+s,0≤i≤d,θi∗=θ0∗+h∗ii+1+s∗,0≤i≤d,φj=hh∗jj−d−1j+r1j+r2,1≤j≤d,ϕj=hh∗jj−d−1j+s∗−r1j+s∗−r2,1≤j≤d.Then, Φ=θii=0d,θi∗i=0d;φjj=1d,ϕjj=1d is a parameter array of Racah type. We refer to the scalars r1, r2, s, s∗, h, h∗, θ0, and θ0∗ as hypergeometric parameters of Φ.Theorem 7 (see [18], Example 5.11).
Fix nonzeros, h∗∈ℱ and some s∗, r, θ0, θ0∗∈ℱ such that neither of r, s∗−r is equal to −i for 1≤i≤d and that s∗ is not equal −i for 2≤i≤2d. Let(15)θi=θ0+si,0≤i≤d,θi∗=θ0∗+h∗ii+1+s∗,0≤i≤d,φj=h∗sjj−d−1j+r,1≤j≤d,ϕj=−h∗sjj−d−1j+s∗−r,1≤j≤d.
Then,Φ=θii=0d,θi∗i=0d;φjj=1d,ϕjj=1d is a parameter array of Hahn type. We refer to the scalars r, s, s∗, h∗, θ0, and θ0∗ as hypergeometric parameters of Φ.Theorem 8 (see [18], Example 5.12).
Fix nonzeroh, s∗∈ℱ and s, r, θ0, θ0∗∈ℱ such that neither of r, s−r is equal to −i for 1≤i≤d, and that s is not equal −i for 2≤i≤2d. Let(16)θi=θ0+hii+1+s,0≤i≤d,θi∗=θ0∗+s∗i,0≤i≤d,φj=hs∗jj−d−1j+r,1≤j≤d,ϕj=hs∗jj−d−1j+r−s−d−1,1≤j≤d.
Then,Φ=θii=0d,θi∗i=0d;φjj=1d,ϕjj=1d is a parameter array of dual Hahn type. We refer to the scalars r, s, s∗, h, θ0, and θ0∗ as hypergeometric parameters of Φ.Theorem 9 (see [18].
Example 5.13). Fix nonzeror, s, s∗∈ℱ, and some θ0, θ0∗∈ℱ such that r≠ss∗:(17)θi=θ0+si,0≤i≤d,θi∗=θ0∗+s∗i,0≤i≤d,φj=rjj−d−1,1≤j≤d,ϕj=r−ss∗jj−d−1,1≤j≤d.
Then,Φ=θii=0d,θi∗i=0d;φjj=1d,ϕjj=1d is a parameter array of Krawtchouk type. We refer to the scalars r, s, s∗, θ0, and θ0∗ as hypergeometric parameters of Φ.


## 3. The Lie Algebrasl2

In this section, we recall some facts concerning the Lie algebrasl2.Definition 5.
(see [11]). The Lie algebra sl2 is the ℱ-algebra that has a basis e,f,h satisfying the following conditions:(18)h,e=2e,h,f=−2f,e,f=h,where −,− denotes the Lie bracket.Lemma 1 (see [10]).
With reference to Definition5, let(19)x=2e−h,y=−2f−h,z=h.Then, x,y,z is a basis for sl2 and(20)x,y=2x+2y,y,z=2y+2z,z,x=2z+2x.
We callx, y, z the equitable basis for the Lie algebra sl2.Lemma 2 (see [11]).
For each nonnegative integerd, there is an irreducible finite-dimensional sl2-module Vd with basis v0, v1, …, vd and action hvi=d−2ivi0≤i≤d, fvi=i+1vi+10≤i≤d−, fvd=0, ev0=0, and evi=d−i+1vi−11≤i≤d. Moreover, up to isomorphism, Vd is the unique irreducible sl2-module of dimension d+1.Lemma 3.
(see [10]). With reference to Lemmas 1 and 2,(21)x+dIv0=0,x+d−2iIvi=2d−i+1vi−1,1≤i≤d,y+d−2iIvi=−2i+1vi+1,0≤i≤d−1,y−dIvd=0,zvi=d−2ivi,0≤i≤d.Definition 6. With reference to Lemmas.
1 and 2, fix ζ∈ℭ. A basis b=b0,b1,…,bd of Vd is said to be a standardζz-eigenbasis whenever ζx, ζy, and ζz act as(22)ζx+dIb0=0,ζx+d−2iIbi=2d−i+1bi−1,1≤i≤d,ζy+d−2iIbi=−2i+1bi+1,0≤i≤d−1,ζy−dIbd=0,ζz−d−2iIbi=0,0≤i≤d.Lemma 4 (see [19]).
With reference to Lemmas1 and 2,(i)
For0≤i≤d, let ui=∑j=0d−i−1id−j!/i!d−j−i!vj. Then, u=u0,u1,…,ud is a standard x-eigenbasis of Vd.(ii)
For0≤i≤d, let wi=∑j=d−id−1ij!/d−i!j+i−d!vj. Then, w=w0,w1,…,wd is a standard y-eigenbasis of Vd.(iii)
Letv0, v1, …, vd be as in Lemma 2. Then, v=v0,v1,…,vd is a standard z-eigenbasis of Vd [10].
The main results of this paper are the following theorems.Theorem 10.
Fix an integer.d≥3; let x, y, and z be the equitable basis of sl2 over ℱ. Let V denote an irreducible sl2-module with dimension d+1; let A∈EndV. Then, the following are equivalent:(i)
Any two of the pairsA,x, A,y, and A,z are Leonard pairs(ii)
All the pairsA,x, A,y, and A,z are Leonard pairsTheorem 11.
Fix an integerd≥3; let x, y, and z be the equitable basis of sl2 over ℱ. Let V denote an irreducible sl2-module with dimension d+1; let A∈EndV such that each of the pairs A,x, A,y, and A,z is a Leonard pair; then, all the pairs are of Krawtchouk type. Moreover, A is a linear combination of 1, x, y, and z.
We remark here that the authors in [12] proved similar result with the generators of the quantum algebra Uqsl2.


## 4. The Type of Leonard Pair

We start recalling some facts which help us to determine the type of the Leonard pairs.Lemma 5 (see [17]).
Letd denote a nonnegative integer, and let V denote a vector space over ℱ with dimension d+1. Let A, A∗ denote a Leonard pair on V with fundamental parameter β. Let the scalars θi and θi∗ be as in Definition 2; then, there exist scalars γ, γ∗, ϱ, and ϱ∗ such that(i)
γ=θi−1−βθi+θi+1,1≤i≤d−1(ii)
γ∗=θi−1∗−βθi∗+θi+1∗,1≤i≤d−1(iii)
ϱ=θi−12−βθi−1θi+θi2−γθi−1+θi,1≤i≤d(iv)
ϱ∗=θi−1∗2−βθi−1∗θi∗+θi∗2−γ∗θi−1∗+θi∗,1≤i≤dTheorem 12.
(see [17]). Let d denote a nonnegative integer, and let V denote a vector space over ℱ with dimension d+1. Let A, A∗ denote a Leonard pair on V. Let β, γ, γ∗, ϱ, ρ∗, ω, η, and η∗ denote a sequence of scalars taken from ℱ, which satisfy (12) and (13). Let the scalars θi, θi∗, ai, and ai∗ be as in Definition 2. Then, the following hold:(i)
ω=ai∗θi−θi+1+ai−1∗θi−1−θi−2−γ∗θi+θi−1,1≤i≤d(ii)
ω=aiθi∗−θi+1∗+ai−1θi−1∗−θi−2∗−γθi∗+θi−1∗,1≤i≤d(iii)
η=ai∗θi−θi−1θi−θi+1−γ∗θi2−ωθi,0≤i≤d(iv)
η∗=aiθi∗−θi−1∗θi∗−θi+1∗−γθi∗2−ωθi∗,0≤i≤dLemma 6.
With reference to Lemma1, if each of the pairs A,x, A,y, and A,z is a Leonard pair, then all the pairs are of dual Hahn type or all the pairs of Krawtchouk type.Proof.
Note that, from Definition4 and Lemma 3, all the pairs are of classical type. Now, the result holds by Theorem 5 and Theorem 6–Theorem 9.
We have two cases to check, the first case, is thereA such that all the pairs A,x, A,y, and A,z are Leonard pairs of dual Hahn type?, and the second case, is there A such that all the pairs A,x, A,y, and A,z are Leonard pairs of Krawtchouk type? We start with the dual Hahn case.
For the rest of the paper, fix an integerd≥3; let Vd denote an irreducible sl2-module with dimension d+1; let A∈EndVd; Let Tb be the matrix that represents the linear map T with respect to the basis b.Lemma 7.
With reference to Lemma3, let ξ∈x,y,z, assume that A, ξ is a Leonard pair of dual Hahn type; let (θii=0d, θi∗i=0d; φjj=1d, ϕjj=1d) be the parameter array associated with the pair A, ξ; then,(23)θi=θ0+hii+1+s,0≤i≤d,θi∗=θ0∗+hξi,0≤i≤d,φi=hhξii−d−1i+rξ,1≤i≤d,ϕi=hhξii−d−1i+rξ−s−d−1,1≤i≤d,where h and hξ are nonzero scalars, none of rξ, s−rξ equal to −i for 1≤i≤d, and s≠i for 2≤i≤2d.Proof.
Clear from Theorem8.Lemma 8.
LetA∈EndVd such that A, z is a Leonard pair of dual Hahn type; then, the basis eii=0d for Vd with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing z is diagonal is kivii=0d or kd−ivd−ii=0d for some nonzero scalars ki∈ℱ.Proof.
Clear from Lemma3 and the paragraph after Definition 2.
Since the case with basiskd−ivd−ii=0d can be treated similar to the case with basis kivii=0d, we shall prove our results only for the case kivii=0d. Let ei=kivii=0d.Lemma 9.
With reference to Lemmas2 and 8,(24)x+dIe0=0,x+d−2iIei=2d−i+1kiki−1ei−1,1≤i≤d,y+d−2iIei=−2i+1kiki+1ei+1,0≤i≤d−1,y−dIed=0,zei=d−2iei,0≤i≤d.Lemma 10.
With reference to Lemmas7 and 8, let A, z be a Leonard pair of dual Hahn type; then,(25)Aei=βi−1∗ei−1+αi∗ei+λi+1∗ei+1,1≤i≤d−1,Ae0=α0∗e0+λ1∗e1,Aed=βd−1∗ed−1+αd∗ed,where(26)βi∗=hd−ii+1+rz,0≤i≤d−1,αi∗=θ0+his+d−2irz+2id−i+d,0≤i≤d,λi∗=hii−1−d+rz+s,1≤i≤d.Proof.
The result is held by Theorem3 and Lemmas 7 and 9.Lemma 11.
With reference to Lemma5 and Theorem 12, let ξ∈x,y,z; assume A, ξ is a Leonard pair of dual Hahn type; let θi=0 and θi∗=0; then,(27)γ=2h,γ∗=0,ϱ=sh22+s,ϱ∗=hξ2,ω=hhξ2rξ−s−2d,η=−dh2hξs1+rξ,η∗=−dhhξ21+rξ.Proof.
Clear from Lemma7.Lemma 12.
LetA∈EndVd; assume A, z is a Leonard pair of dual Hahn type; then, the pair A, y is not a Leonard pair of dual Hahn type.Proof.
Let(28)H=A∗2A−βA∗AA∗+AA∗2−γ∗A∗A+AA∗−ϱ∗A−γA∗2+ωA∗+η∗I.
Since the pairA, z is a Leonard pair, the action of A on the basis e=eii=0d is given in Lemma 10, and the action of y is given in Lemma 9. Assume that the pair A, y is a Leonard pair and let y=A∗; hence, by Theorem 4, there exists a sequence of scalars β, γ, γ∗, ϱ, ρ∗, ω, η, and η∗ as in Lemma 11 such that He=0. Now, if we solve He1,1 for ry, we find that ry=k1rz−k0rz+1/k1. Substitute ry; then, for 1≤i≤d−2, solve the i+1,i-entries of He for ki; we find that ki=ki+1rz+i+2/ki+2rz+i+1. Substitute ki to find that(29)He4,1=48kd−12hrz+d2kdrz+1rz+2rz+3.
By Lemmas7 and 8, h≠0, rz+d≠0, and kd−1≠0, so He4,1≠0. Hence, the pair A, y is not a Leonard pair.Lemma 13.
LetA, z be a Leonard pair of dual Hahn type; then, the pair A, x is not a Leonard pair of dual Hahn type.Proof.
Similar to proof of Lemma12.


## 5. Leonard Pair of Krawtchouk Type

In this section, we describe a linear mapA such that all the pairs A,x, A,y, and A,z are Leonard pairs of Krawchouch type.Lemma 14.
With reference to Lemma3, let ξ∈x,y,z, assume that A, ξ is a Leonard pair of Krawtchouk type; let (θii=0d, θi∗i=0d; φjj=1d, ϕjj=1d) be the parameter array associated with the pair A, ξ; then, there exist a nonzero scalars h, hξ, and rξ such that rξ−hhξ≠0, and(30)θi=θ0+hi,0≤i≤d,θi∗=θ0∗+hξi,0≤i≤d,φi=rξii−d−1,1≤i≤d,ϕi=rξ−hhξii−d−1,1≤i≤d,Proof.
Clear from Theorem9.Lemma 15.
With reference to Lemma5 and Theorem 12, assume A, ξ is a Leonard pair of Krawtchouk type; let θi=0 and θi∗=0; then,(31)γ=0,γ∗=0,ϱ=h2,ϱ∗=hξ,ω=−hhξ−2rξ,η=−dhrξ,η∗=−dhξrξ.Proof.
Clear from Lemma14.Lemma 16.
With reference to Lemmas14 and 8, Let A, z be a Leonard pair of Krawtchouk type; then, Aei=βi−1ei−1+αiei+λi+1ei+11≤i≤d−1, Ae0=α0e0+λ1e1, and Aed=βd−1ed−1+αded, where(32)βi=d−irz2,0≤i≤d−1,αi=θ0+ih−d−2irz2,0≤i≤d,λi=−irz+2h2,1≤i≤d.Proof.
The result holds by Theorem3 and Lemmas 3 and 14.Lemma 17.
With reference to Lemma16, the pair Ae, ze is a Leonard pair of Krawtchouk type in Matd+1ℱ if and only if h≠0, rz≠0, and rz+2h≠0.Proof.
LetP be d+1×d+1 matrix indexed 1,2,…,d+1 such that the i,j-entry is(33)Pij=−2j−1i−1!i−j!,j≤i,0,j>i.
Then, the matrices representP−1AeP and P−1zeP as in (1), where θii=0d, θi∗i=0d; φii=1d and ϕii=1d are as in Lemma 14. Hence, by Theorem 2, P−1AeP, P−1zeP is a Leonard pair of Krawtchouk type if and only if h≠0, rz≠0, and rz+2h≠0 hold, which implies that the pair Ae, ze is a Leonard pair of Krawtchouk type if and only if h≠0, rz≠0, and rz+2h≠0.Lemma 18.
With reference to Lemmas8 and 14, assume Ae, ze is a Leonard pair of Krawtchouk type. If the pair Ae, ye is a Leonard pair, then there exists a nonzero scalar t∈ℱ such that ki/ki−1=t1≤i≤d. Moreover, ry=rz1−t/t.Proof.
Routine calculations using Theorem4 and Lemma 15.Lemma 19.
With reference to Lemmas8 and 14, assume Ae, ze is a Leonard pair of Krawtchouk type. If the pair Ae, xe is a Leonard pair, then there exists a nonzero scalar t∈ℱ such that ki/ki−1=t1≤i≤d. Moreover, rx=trz+2h−rz.Proof.
Routine calculations using Theorem4 and Lemma 15.Lemma 20.
Suppose that the pairsA, z and A, y are Leonard pairs of Krawtchouk type; let θii=0d, θi∗i=0d and φii=1d, ϕii=1d be the parameter array associated with the pair A, y. Then, there exist a nonzero scalars h, rz, and t such that t≠1, rz+2h≠0, rz−trz+2h≠0, and(34)θi=θ0+hi,0≤i≤d,θi∗=2i−d,0≤i≤d,φi=rz1−tii−d−1t,1≤i≤d,ϕi=rz−trz+2hii−d−1t,1≤i≤d.Proof.
Letk0=1 in Lemma 8, so the basis e=tivii=0d. Hence, the result holds by Lemmas 14 and 18.Lemma 21.
Suppose that the pairsA, z and A, x are Leonard pairs of Krawtchouk type, let θii=0d, θi∗i=0d and φii=1d, ϕii=1d be the parameter array associated with the pair A, x. Then, there exist nonzero scalars h, rz, and t such that t≠1, rz+2h≠0, rz−trz+2h≠0, and(35)θi=θ0+hi,0≤i≤d,θi∗=2i−d,0≤i≤d,φi=trz+2h−rzii−d−1,1≤i≤d,ϕi=t−1rz+2hii−d−1,1≤i≤d.Proof.
Letk0=1 in Lemma 8, so the basis e=tivii=0d. Hence, the result holds by Lemmas 14 and 19.Lemma 22.
With reference to Lemma2, let t∈ℱ be a nonzero scalar; let e=tivii=0d. Let A be as in Lemma 16; assume that Ae, ze is a Leonard pair of Krawtchouk type; then, the pair Ae, ye is a Leonard pair of Krawtchouk type if and only if t≠1 and rz−trz+2h≠0.Proof.
The pairAe, ze is a Leonard pair of Krawtchouk type; then, by Lemma 17, h≠0, rz≠0, and rz+2h≠0.
LetP be d+1×d+1 matrix indexed 1,2,…,d+1 such that the i,j-entry is(36)Pij=2j−1t−1j−1i−1!tj−1i−j!,j≤i,0,j>i.
Then, the matrices representP−1AeP, P−1yeP as in (1), where θii=0d, θi∗i=0d and φii=1d, ϕii=1d are as in Lemma 20. Hence, by Theorem 2, P−1AeP, P−1yeP is a Leonard pair of Krawtchouk type, if and only if t≠1 and rz−trz+2h≠0 hold, which implies that the pair Ae, ye is a Leonard pair of Krawtchouk type if and only if t≠1 and rz−trz+2h≠0.Lemma 23.
With reference to Lemma2, let t∈ℱ be a nonzero scalar; let e=tivii=0d. Let A be as in Lemma 16; assume, that Ae, ze is a Leonard pair of Krawtchouk type; then ,the pair Ae, xe is a Leonard pair of Krawtchouk type if and only if t≠1 and rz−trz+2h≠0.Proof.
The pairAe, ze is a Leonard pair of Krawtchouk type; then, by Lemma 17, h≠0, rz≠0, and rz+2h≠0.
LetP be d+1×d+1 matrix indexed 1,2,…,d+1 such that the i,j-entry is(37)Pij=2j−d−1t−1j−1d−i+1!tdd!d−i−j+!,j≤d−i+2,0,j>d−i+2.
Then,(38)P−1xeP=θd1θd−11θd−2⋱⋱1θ0,P−1AeP=θ0∗ϕ1θ1∗ϕ2θ2∗⋱⋱ϕdθd∗,where θii=0d, θi∗i=0d and φii=1d, ϕii=1d are as in Lemma 21. Let G=T−1KT∗, where T, K, and T∗ are in Matd+1ℱ such that Ti+1,j+1=Πh=0j−1θi∗−θh∗, Ti+1,j+1∗=Πh=0j−1θd−i∗−θd−h∗ for 0≤i,j≤d, and Kij=1 if i+j=d+2 and 0 otherwise. Then, the matrices represent GP−1AePG−1, GP−1xePG−1 as in (1). The rest of the proof will be similar to proof of Lemma 22.
We remark here that the existence ofG that appears in proof of Lemma 23 was proved by Terwilliger in [18].Lemma 24.
With reference to Lemma2, let t∈ℱ be a nonzero scalar; let e=tivii=0d. Let A be as in Lemma 16, assume that Ae, ze is a Leonard pair of Krawtchouk type; then, Ae, xe is a Leonard pair of Krawtchouk type if and only if Ae, ye is a Leonard pair of Krawtchouk type.Proof.
Clear from Lemmas22 and 23.Lemma 25.
Fix an integerd≥3, let t∈ℱ be a nonzero scalar; let x, y, z denote the equitable basis for the Lie algebra sl2. Let Vd denote a finite-dimensional irreducible sl2-module, let vii=0d be a basis of Vd as in Lemma 3, let e=tivii=0d, and let A∈EndVd such that A acts on e as in Lemma 16. Then, each of the pairs Ae,xe, Ae,ye, Ae,ze is a Leonard pair of Krawtchouk type if and only if rz≠0, h≠0, t≠1, rz+2h≠0 and rz−trz+2h≠0.Proof.
The result holds from Lemmas17, 22, and 23.


## 6. A Basis ofA

LetUsl2 denote the universal enveloping algebra of sl2. Thus, Usl2 is the associative ℱ-algebra with generators x, y, z and relations x,y=2x+2y, y,z=2y+2z, and z,x=2z+2x, where n,m=nm−mn is commutator of n and m. In this section, we write A as a linear combination of the generators of Usl2 [16, 20–23].Definition 7.
LetY∈Usl2 denote any linear combination of 1, x, y, and z, and write(39)Y=aII+axx+ayy+azz,where aI, ax, ay, az∈ℱ.Lemma 26.
With reference to Lemma3 and Definition 7, let t be a nonzero scalar in ℱ and let ei=tivi0≤i≤d; then,(40)Ye0=κ0e0+η1e1,Yed=μd−1ed−1+κded,Yei=μi−1ei−1+κiei+ηi+1ei+1,1≤i≤d−1,where(41)κi=aI+d−2iaz−ax−ay,0≤i≤d,μi=2d−itax,0≤i≤d−1,ηi=−2iayt,1≤i≤d.Proof.
Expand the action ofx, y, and z on e=eii=0d using Lemma 8.Lemma 27.
Fix an integerd≥3; let x, y, z denote the generators of Usl2, let A∈Usl2 such that A=aII+axx+ayy+azz, let Vd denote a finite-dimensional irreducible sl2-module, and let e=tivii=0d be a basis of Vd, assume that A acts on e as in Lemma 16. Then,(42)aI=θ0+dh2,ax=rz4t,ay=trz+2h4,az=trz+2h4−rz+h2+rz4t,rz=4tax,h=2tay−t2ax.Proof.
Compare the action ofA on e with the action of Y in Lemma 26 to get the result.Lemma 28.
With reference to Lemma27, az=t1−ttax−ay.Proof.
Routine calculations show that(43)t1−ttax−ay=trz+2h4−rz+h2+rz4t.Lemma 29.
Fix an integerd≥3; let x, y, z denote the generators of Usl2, let A∈Usl2 such that A=aII+axx+ayy+azz, where aI,ax, ay, and az in ℱ, and let Vd denote a finite-dimensional irreducible sl2-module. Then, each of the pairs A,x, A,y, and A,z is a Leonard pair of Krawtchouk type if and only if ax, ay, az are nonzero scalars in ℱ and there exists a nonzero scalar t∈ℱ such that az=t1−ttax−ay and ay−t2ax≠0.Proof.
Assume all the pairsA,x, A,y, and A,z are Leonard pairs of Krawtchouk type; then, by Lemma 25, there exist nonzero scalars rz, h, t∈ℱ, t≠1 such that A acts on the basis tivii=0d as in Lemma 16. Moreover, rz+2h≠0 and rz−trz+2h≠0. Hence, A acts on tivii=0d as aII+axx+ayy+azz, where aI,ax, ay, and az are as in Lemma 27, and az=t1−ttax−ay.
Note thatrz, rz+2h, and h are nonzero scalars if and only if ax, ay, and ay−t2ax are nonzero scalars, respectively, and since rz−trz+2h=4tax−ay, then, by Lemma 28, rz−trz+2h≠0 if and only if az≠0.
In the another direction, ifA=aII+axx+ayy+azz for aI∈ℱ, ax, ay, az are nonzero scalars in ℱ, az=t1−ttax−ay and ay−t2ax≠0 for nonzero scalar t∈ℱ, then az≠0 implies t≠1; hence, we can find rz=4tax, h=2/tay−t2ax nonzero scalars such that A acts on tivii=0d as in Lemma 16. Hence, the pairs A,x, A,y, and A,z are Leonard pairs of Krawtchouk type by Lemma 25.Proof of Theorem 10.
Note that, from Lemma4, xu=yw=zv. Hence, the result holds from Lemmas 16, 17, and 24.Proof of Theorem 11.
Clear from Lemmas12, 25, and 29.



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