comment out above line \maketitle \begin{abstract} Let $\K$ denote a field and let $V$ denote a vector space over $\K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\rightarrow V$ and $A^*:V\rightarrow V$ that satisfy conditions (i)
(ii) below. \begin{enumerate} em There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. em There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. \end{enumerate} We call such a pair a {Leonard pair} on $V$. We give an overview of the theory of Leonard pairs. \end{abstract} \section{Leonard pairs} \medskip %\noindent Throughout $\K$ will denote an arbitrary %field. \medskip \noindent We begin by recalling the notion of a Leonard pair. We will use the following terms. Let $X$ denote a square matrix. Then $X$ is called {tridiagonal} whenever each nonzero entry lies on either the diagonal
the subdiagonal
or the superdiagonal. Assume $X$ is tridiagonal. Then $X$ is called {irreducible} whenever each entry on the subdiagonal is nonzero and each entry on the superdiagonal is nonzero. \medskip \noindent We now define a Leonard pair. For the rest of this paper $\K$ will denote a field. \begin{definition} \label{def:lprecall} \cite{LS99} Let $V$ denote a vector space over $\K$ with finite positive dimension. By a {Leonard pair} on $V$ we mean an ordered pair of linear transformations $A:V\rightarrow V$ and $A^*:V\rightarrow V$ that satisfy conditions (i)
(ii) below. \begin{enumerate} em There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. em There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. \end{enumerate} \end{definition} \begin{note} According to a common notational convention
$A^*$ denotes the conjugate transpose of $A$. We are not using this convention. In a Leonard pair $A
A^*$ the linear transformations $A
A^*$ are arbitrary subject to (i)
(ii) above. \end{note} \noindent \begin{note} Our use of the name ``Leonard pair'' is motivated by a connection to a theorem of D. Leonard \cite[p. 260]{BanIto}
\cite{Leon} involving the $q$-Racah and related polynomials of the Askey Scheme. %See %\cite[App. A]{LS99} and %\cite{LS24} for more information %on this. \end{note} %\begin{note} According to a common notational convention
%for a linear transformation $A$ the conjugate-transpose of $A$ %is denoted $A^*$. We emphasize we are not using this convention. %In a Leonard pair $A
A^*$
the linear transformations $A
A*$ %are arbitrary subject to (i)
(ii) above. %\end{note} \section{An example of a Leonard pair} \medskip \noindent Here is an example of a Leonard pair. Set $V={\K}^4$ (column vectors)
set \beast A = \left( \begin{array}{ c c c c } 0 & 3 & 0 & 0
1 & 0 & 2 & 0
0 & 2 & 0 & 1
0 & 0 & 3 & 0
\end{array} \right)
\qquad A^* = \left( \begin{array}{ c c c c } 3 & 0 & 0 & 0
0 & 1 & 0 & 0
0 & 0 & -1 & 0
0 & 0 & 0 & -3
\eeast and view $A$ and $A^*$ as linear transformations on $V$. We assume the characteristic of $\K$ is not 2 or 3 to ensure $A$ is irreducible. Then the pair $A
A^*$ is a Leonard pair on $V$. Indeed condition (i) in Definition \ref{def:lprecall} is satisfied by the basis for $V$ consisting of the columns of the 4 by 4 identity matrix. To verify condition (ii)
we display an invertible matrix $P$ such that $P^{-1}AP$ is diagonal and $P^{-1}A^*P$ is irreducible tridiagonal. Set \beast P = \left( \begin{array}{ c c c c} 1 & 3 & 3 & 1
1 & 1 & -1 & -1
1 & -1 & -1 & 1
1 & -3 & 3 & -1
\end{array} \right). \eeast By matrix multiplication $P^2=8I$
where $I$ denotes the identity
so $P^{-1}$ exists. Also by matrix multiplication
\begin{equation} AP = PA^*. \label{eq:apeq} \end{equation} Apparently $P^{-1}AP$ is equal to $A^*$ and is therefore diagonal. By (\ref{eq:apeq}) and since $P^{-1}$ is a scalar multiple of $P$
we find $P^{-1}A^*P$ is equal to $A$ and is therefore irreducible tridiagonal. Now condition (ii) of Definition \ref{def:lprecall} is satisfied by the basis for $V$ consisting of the columns of $P$. \section{Leonard systems} When working with a Leonard pair
it is often convenient to consider a closely related and somewhat more abstract concept called a {Leonard system}. In order to define this we recall a few terms. Let $V$ denote a vector space over $\K$ with finite positive dimension. Let $\mbox{End}(V)$ denote the $\K$-algebra consisting of the linear transformations from $V$ to $V$. For $A \in \mbox{End}(V)$
by the {eigenvalues} of $A$ we mean the roots of the characteristic polynomial of $A$. These eigenvalues are contained in the algebraic closure of $\K$. We say $A$ is {multiplicity-free} whenever the eigenvalues of $A$ are mutually distinct and contained in $\K$. Assume for the moment that $A$ is multiplicity-free. Let $\theta_0
\theta_1
\ldots
\theta_d$ denote an ordering of the eigenvalues of $A$. For $0 \leq i \leq d$ let $v_i$ denote a nonzero vector in $V$ that is an eigenvector for $A$ with eigenvalue $\theta_i$. Observe the sequence $v_0
v_1
v_d$ is a basis for $V$. For $0 \leq i \leq d$ define $E_i \in \mbox{End}(V)$ so that $E_iv_j=\delta_{ij}v_j$ for $0 \leq j\leq d$. We call $E_i$ the {primitive idempotent} of $A$ associated with $\theta_i$. \begin{definition} \label{def:ls} \cite{LS99} Let $d$ denote a nonnegative integer and let $V$ denote a vector space over $\K$ with dimension $d+1$. By a {Leonard system} on $V$ we mean a sequence $\Phi=(A;A^*; $ $\lbrace E_i \rbrace_{i=0}^d; \lbrace E^*_i \rbrace_{i=0}^d)$ that satisfies conditions (i)--(v) below. \begin{enumerate} em Each of $A
A^*$ is a multiplicity-free element of $\mbox{End}(V)$. em $E_0
E_1
E_d$ is an ordering of the primitive idempotents of $A$. em $E^*_0
E^*_1
E^*_d$ is an ordering of the primitive idempotents of $A^*$. em ${\displaystyle{ E^*_iAE^*_j = \cases{0, &if $\;|i-j|> 1$;\cr \not=0, &if $\;|i-j|=1$\cr} \qquad \qquad (0 \leq i,j\leq d). }}$ em ${\displaystyle{ E_iA^*E_j = \cases{0, &if $\;|i-j|> 1$;\cr \not=0, &if $\;|i-j|=1$\cr} \qquad \qquad (0 \leq i,j\leq d). }}$ \end{enumerate} We call $d$ the {diameter} of $\Phi$. \end{definition} Leonard pairs and Leonard systems are related as follows. \begin{theorem} \cite{TLT:split} For $A
A^*$ in $\mbox{End}(V)$
the pair $A
A^*$ is a Leonard pair on $V$ if and only if the following (i)
(ii) hold. \begin{enumerate} em Each of $A
A^*$ is multiplicity-free. em There exists an ordering $E_0
E_d$ of the primitive idempotents of $A$ and there exists an ordering $E^*_0
E^*_d$ of the primitive idempotents of $A^*$ such that $(A;A^*;$ $\lbrace E_i \rbrace_{i=0}^d; \lbrace E^*_i \rbrace_{i=0}^d)$ is a Leonard system on $V$. \end{enumerate} \end{theorem} \section{Leonard pairs from 24 points of view} \medskip Let $A
A^*$ denote a Leonard pair on $V$. We describe 24 bases for $V$ on which $A
A^*$ act in an attractive fashion. \medskip \noindent To describe the bases we will use the following terms. Let $X$ denote a square matrix. Then $X$ is called {lower bidiagonal} whenever each nonzero entry lies on either the diagonal or the subdiagonal. We say $X$ is {upper bidiagonal} whenever the transpose of $X$ is lower bidiagonal. Let $v_0
v_d$ denote a basis for $V$. By the {inversion} of this basis we mean the basis $v_d
v_{d-1}
v_0$. \medskip \noindent The 24 bases are described by the diagram below. In that diagram each vertex represents one of the 24 bases. The shading on the vertex indicates the nature of the $A
A^*$ action. \begin{enumerate} em Black: $A$ is diagonal and $A^*$ is irreducible tridiagonal. em Green: $A$ is lower bidiagonal and $A^*$ is upper bidiagonal. em Red: $A$ is upper bidiagonal and $A^*$ is lower bidiagonal. em Yellow: $A$ is irreducible tridiagonal and $A^*$ is diagonal. \end{enumerate} \noindent For each pair of bases in the diagram that are connected by an arc
consider the transition matrix from one of these bases to the other. The shading on the arc indicates the nature of this transition matrix. \begin{enumerate} em Solid arc: Transition matrix is diagonal. em Dashed arc: Transition matrix is lower triangular. em Dotted arc: The bases are the inversion of one another. \end{enumerate} \bigskip \input psfig.sty \centerline{\psfig{figure=s4oct1.eps,height=4.5cm}} \vskip .5cm \bigskip \noindent See \cite{LS24} for more information concerning the 24 bases. \section{The classifying space} We will shortly give a classification of the Leonard systems. In order to describe the result we recall the notion of a {parameter array}. \begin{definition} %\cite[Definition 10.1]{conform} \label{def:pa} Let $d$ denote a nonnegative integer. By a {parameter array over $\K$ of diameter $d$} we mean a sequence of scalars $ (\theta_i
\theta^*_i
i=0..d; \varphi_j
\phi_j
j=1..d) $ taken from $\K$ which satisfy the following conditions (PA1)--(PA5). \begin{description} em[(PA1)] $ \theta_i \not=\theta_j
\qquad \theta^*_i \not=\theta^*_j \qquad \mbox{if}\quad i\not=j
\qquad \qquad (0 \leq i
j\leq d) $.
em[(PA2)] $ \varphi_i \not=0
\qquad \phi_i \not=0 \qquad \qquad (1 \leq i \leq d). $
em[(PA3)] $\varphi_i = \phi_1 \sum_{h=0}^{i-1} \frac{\theta_h-\theta_{d-h}}{\theta_0-\theta_d} + (\theta^*_i-\theta^*_0)(\theta_{i-1}-\theta_d) \qquad \qquad (1 \leq i \leq d)$.
em[(PA4)] $\phi_i = \varphi_1 \sum_{h=0}^{i-1} \frac{\theta_h-\theta_{d-h}}{\theta_0-\theta_d} + (\theta^*_i-\theta^*_0)(\theta_{d-i+1}-\theta_0) \qquad \qquad (1 \leq i \leq d)$.
em[(PA5)] The expressions \begin{eqnarray*} \frac{\theta_{i-2}-\theta_{i+1}}{\theta_{i-1}-\theta_i}
\qquad \frac{\theta^*_{i-2}-\theta^*_{i+1}}{\theta^*_{i-1}-\theta^*_i} \label{eq:betaplusone} \end{eqnarray*} are equal and independent of $i$ for $2 \leq i \leq d-1$. \end{description} \end{definition} \section{The classification of Leonard systems} In this section we give a bijection from the set of parameter arrays to the set of isomorphism classes of Leonard systems. Let $(\theta_i
i=0..d;\varphi_j
j=1..d)$ denote a parameter array over $\K$. Define \begin{eqnarray*} A=\left( \begin{array}{c c c c c c} \theta_0 & & & & & {\mathbf 0}
1 & \theta_{1} & & & &
& 1 & \theta_{2} & & &
& & \cdot & \cdot & &
& & & \cdot & \cdot &
{\mathbf 0}& & & & 1 & \theta_d \end{array} \right)
\quad A^*=\left( \begin{array}{c c c c c c} \theta^*_0 &\varphi_1 & & & & {\mathbf 0}
& \theta^*_1 & \varphi_2 & & &
& & \theta^*_2 & \cdot & &
& & & & \cdot & \varphi_d
{\mathbf 0}& & & & & \theta^*_d \end{array} \right). \end{eqnarray*} Observe $A$ (resp. $A^*$) is multiplicity free with eigenvalues $\theta_0
\theta_d$ (resp. $\theta^*_0
\theta^*_1
\theta^*_d$). For $0 \leq i \leq d$ let $E_i$ (resp. $E^*_i$) denote the primitive idempotent of $A$ (resp. $A^*$) associated with $\theta_i$ (resp. $\theta^*_i$). Then the sequence $(A;A^*; $ $\lbrace E_i \rbrace_{i=0}^d; \lbrace E^*_i \rbrace_{i=0}^d)$ is a Leonard system on $\K^{d+1}$. This construction induces a bijection from the set of parameter arrays to the set of isomorphism classes of Leonard systems \cite{LS99}. \section{Leonard pairs $A,A^*$ with $A$ lower bidiagonal and $A^*$ upper bidiagonal} \noindent Let $A
A^*$ denote matrices in $\hbox{Mat}_{d+1}(\K)$. Let us assume $A$ is lower bidiagonal and $A^*$ is upper bidiagonal. We give a necessary and sufficient condition for $A
A^*$ to be a Leonard pair. \begin{theorem} \cite{conform} \label{thm:lug} Let $d$ denote a nonnegative integer and let $A
A^*$ denote matrices in $\hbox{Mat}_{d+1}(\K)$. Assume $A$ lower bidiagonal and $A^*$ is upper bidiagonal. Then the following (i)
(ii) are equivalent. \begin{enumerate} em The pair $A
A^*$ is a Leonard pair. em There exists a parameter array $(\theta_i
j=1..d)$ over $\K$ such that \begin{eqnarray*} &&A_{ii} =\theta_i
\qquad \qquad A^*_{ii} =\theta^*_i \qquad \qquad (0 \leq i \leq d)
\label{eq:comb1}
&&\qquad A_{i,i-1}A^*_{i-1,i} = \varphi_i \qquad \qquad (1 \leq i \leq d). \label{eq:comb2} \end{eqnarray*} \end{enumerate} \end{theorem} \section{Leonard pairs $A,A^*$ with $A$ tridiagonal and $A^*$ diagonal} \noindent Let $A
A^*$ denote matrices in $\hbox{Mat}_{d+1}(\K)$. Let us assume $A$ is tridiagonal and $A^*$ is diagonal. We give a necessary and sufficient condition for $A
A^*$ to be a Leonard pair. \begin{theorem} \cite{conform} \label{thm:tdcrit} Let $d$ denote a nonnegative integer and let $A
A^*$ denote matrices in $\hbox{Mat}_{d+1}(\K)$. Assume $A$ is tridiagonal and $A^*$ is diagonal. Then the following (i)
j=1..d)$ over $\K$ such that \begin{eqnarray*} A_{ii} &=& \theta_i + \frac{\varphi_i}{\theta^*_i-\theta^*_{i-1}} + \frac{\varphi_{i+1}}{\theta^*_i-\theta^*_{i+1}} \qquad \qquad (0 \leq i \leq d)
\label{eq:aiiform}
A_{i,i-1}A_{i-1,i}&=& \varphi_i\phi_i \frac{\prod_{h=0}^{i-2}(\theta^*_{i-1}-\theta^*_h) } {\prod_{h=0}^{i-1}(\theta^*_{i}-\theta^*_h) } \
\frac{\prod_{h=i+1}^d (\theta^*_i-\theta^*_h) } {\prod_{h=i}^d(\theta^*_{i-1}-\theta^*_h) } \qquad (1 \leq i \leq d)
\label{eq:crossprod}
A^*_{ii} &=&\theta^*_i \qquad \qquad (0 \leq i \leq d). \label{eq:ths} \end{eqnarray*} \end{enumerate} \end{theorem} \section{A characterization of the parameter arrays I} In this section we characterize the parameter arrays in terms of bidiagonal matrices. We will refer to the following set-up. \begin{definition} \label{def:setup} Let $d$ denote a nonnegative integer and let $(\theta_i
j=1..d) $ denote a sequence of scalars taken from $\K$. We assume this sequence satisfies PA1 and PA2. \end{definition} \begin{theorem} \cite{TLT:array} \label{thm:gmat} With reference to Definition \ref{def:setup}
the following (i)
(ii) are equivalent. \begin{enumerate} em The sequence $(\theta_i
j=1..d)$ satisfies PA3--PA5. em There exists an invertible matrix $G \in \mbox{Mat}_{d+1}(\K)$ such that both \begin{eqnarray*} G^{-1} \left( \begin{array}{c c c c c c} \theta_0 & & & & & {\mathbf 0}
{\mathbf 0}& & & & 1 & \theta_d \end{array} \right) G &=&\left( \begin{array}{c c c c c c} \theta_d & & & & & {\mathbf 0}
1 & \theta_{d-1} & & & &
& 1 & \theta_{d-2} & & &
{\mathbf 0}& & & & 1 & \theta_0 \end{array} \right)
\label{eq:g1}
G^{-1} \left( \begin{array}{c c c c c c} \theta^*_0 &\varphi_1 & & & & {\mathbf 0}
{\mathbf 0}& & & & & \theta^*_d \end{array} \right) G &=& \left( \begin{array}{c c c c c c} \theta^*_0 &\phi_1 & & & & {\mathbf 0}
& \theta^*_1 & \phi_2 & & &
& & & & \cdot & \phi_d
{\mathbf 0}& & & & & \theta^*_d \end{array} \right). \label{eq:g2} \end{eqnarray*} \end{enumerate} \end{theorem} \section{A characterization of the parameter arrays II} \noindent In this section we characterize the parameter arrays in terms of polynomials. We will use the following notation. Let $\lambda $ denote an indeterminate
and let $\K \lbrack \lambda \rbrack $ denote the $\K$-algebra consisting of all polynomials in $\lambda $ which have coefficients in $\K$. From now on all polynomials which we discuss are assumed to lie in $\K\lbrack \lambda \rbrack $. \begin{theorem} \cite{TLT:array} \label{eq:mth} With reference to Definition \ref{def:setup}
j=1..d)$ satisfies PA3--PA5. em For $0 \leq i \leq d$ the polynomial \begin{equation} \sum_{n=0}^i \frac{ (\lambda-\theta_0) (\lambda-\theta_1) \cdots (\lambda-\theta_{n-1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\varphi_1\varphi_2\cdots \varphi_n} \label{eq:poly1} \end{equation} is a scalar multiple of the polynomial \begin{eqnarray*} \sum_{n=0}^i \frac{ (\lambda-\theta_d) (\lambda-\theta_{d-1}) \cdots (\lambda-\theta_{d-n+1}) (\theta^*_i-\theta^*_0) (\theta^*_i-\theta^*_1) \cdots (\theta^*_i-\theta^*_{n-1}) } {\phi_1\phi_2\cdots \phi_n}. \label{eq:poly2} \end{eqnarray*} \end{enumerate} \end{theorem} \section{Some orthogonal polynomials of the Askey scheme} There is a natural correspondence between Leonard systems and a class of orthogonal polynomials consisting of the $q$-Racah polynomials and some related polynomials of the Askey scheme. This correspondence is described as follows \cite{TLT:array}. \medskip \noindent Let $\Phi$ %=(A;A^*; $ $\lbrace E_i \rbrace_{i=0}^d; %\lbrace E^*_i \rbrace_{i=0}^d)$ denote a Leonard system. By the {polynomials which correspond to $\Phi$} we mean the polynomials from (\ref{eq:poly1})
where the parameter array from that line is the one associated with $\Phi$. The polynomials which correspond to a Leonard system are listed below: \medskip \noindent $q$-Racah
$q$-Hahn
dual $q$-Hahn
$q$-Krawtchouk
dual $q$-Krawtchouk
quantum $q$-Krawtchouk
affine $q$-Krawtchouk
Racah
Hahn
dual-Hahn
Krawtchouk
Bannai/Ito
and orphan polynomials. \medskip \noindent The Bannai/Ito polynomials can be obtained from the $q$-Racah polynomials by letting $q$ tend to $-1$. The orphan polynomials have maximal degree 3 and exist for $\mbox{char}(\K)=2$ only. \medskip \noindent See \cite{KoeSwa} for information on the Askey scheme. \section{The Askey-Wilson relations} We turn our attention to the representation theoretic aspects of Leonard pairs. \begin{theorem} \cite{aw} \label{lptheorem} Let $V$ denote a vector space over $\K$ with finite positive dimension. Let $A
A^*$ denote a Leonard pair on $V$. Then there exists a sequence of scalars $\beta
\gamma
\gamma^*
\varrho
\varrho^*$
$\omega
\eta
\eta^*$ taken from $\K$ such that both \begin{eqnarray*} \label{askwil1} A^2 A^*-\beta A A^*\!A+A^*\!A^2-\gamma\left( A A^*\!+\!A^*\!A \right)-\varrho\
A^* &=& \gamma^*\!A^2+\omega A+\eta\
I
\label{askwil2} A^*{}^2\!A-\beta A^*\!AA^*\!+AA^*{}^2- \gamma^*\!\left(A^*\!A\!+\!A A^*\right)-\varrho^*\!A &=& \gamma A^*{}^2+\omega A^*\!+\eta^*I. \end{eqnarray*} The sequence is uniquely determined by the pair $A
A^*$ provided the dimension of $V$ is at least $4$. \end{theorem} \noindent The following theorem is a kind of converse to Theorem \ref{lptheorem}. \begin{theorem} \cite{aw} Let $V$ denote a vector space over $\K$ with finite positive dimension. Let $A:V\to V$ and $A^*:V\to V$ denote linear transformations. Suppose that: \begin{itemize} em There exists a sequence of scalars $\beta
\varrho^*
\omega
\eta^*$ taken from $\K$ which satisfies {(\ref{askwil1})}
{(\ref{askwil2})}. em $q$ is not a root of unity
where $q+q^{-1}=\beta$. em Each of $A$ and $A^*$ is multiplicity-free. em There does not exist a subspace $W\subseteq V$ such that $W\not=0$
$W\not=V$
$AW\subseteq W$
$A^*W\subseteq W$. \end{itemize} Then $A
A^*$ is a Leonard pair on $V$. \end{theorem} \section{Leonard pairs and the Lie algebra $sl_2$} \medskip \noindent In this section we assume the field $\K$ is algebraically closed with characteristic zero. \medskip \noindent We recall the Lie algebra $sl_2=sl_2(\K)$. This algebra has a basis $e
f
h$ satisfying \beast &&\lbrack h
e \rbrack = 2e
\qquad \lbrack h
f \rbrack = -2f
\qquad \lbrack e
f \rbrack = h
\eeast where $\lbrack\
\
\rbrack $ denotes the Lie bracket. \medskip \noindent We recall the irreducible finite dimensional modules for $sl_2$. \begin{lemma} \cite[p. 102]{Kassel} \label{lemvd} There exists a family \begin{equation} \label{sl2modlist} V_d \qquad \qquad d = 0
1
2\ldots \end{equation} of irreducible finite dimensional $sl_2$-modules with the following properties. The module $V_d$ has a basis $v_0
v_d$ satisfying $h v_i =(d-2i)v_i$ for $0 \leq i \leq d$
$fv_i = (i+1)v_{i+1}$ for $0 \leq i \leq d-1$
$fv_d=0$
$ev_i = (d-i+1)v_{i-1}$ for $1 \leq i \leq d$
$ev_0=0$. Every irreducible finite dimensional $sl_2$-module is isomorphic to exactly one of the modules in line (\ref{sl2modlist}). \end{lemma} \begin{theorem} \label{ex:slgen} \cite[Ex. 1.5]{TD00} Let $A$ and $A^*$ denote semi-simple elements in $sl_2$ and assume $sl_2$ is generated by these elements. Let $V$ denote an irreducible finite dimensional module for $sl_2$. Then the pair $A
A^*$ acts on $V$ as a Leonard pair. \end{theorem} \noindent We remark the Leonard pairs in Theorem \ref{ex:slgen} correspond to the Krawtchouk polynomials \cite{KoeSwa}. \section{Leonard pairs and $U_q(sl_2)$} \medskip \noindent In this section we assume $\K$ is algebraically closed. We fix a nonzero scalar $q\in \K$ which is not a root of unity. We recall the quantum algebra $U_q(sl_2)$. \begin{definition} \cite[p.122]{Kassel} \label{def:uqsl2} Let $U_{q}(sl_2)$ denote the associative $\K$-algebra with 1 generated by symbols $ e
k
k^{-1}$ subject to the relations \beast kk^{-1} = k^{-1}k= 1
\eeast \beast ke = q^2 ek
\qquad\qquad kf = q^{-2}fk
\eeast \beast ef - fe = {{k-k^{-1}}\over {q - q^{-1}}}. \eeast \end{definition} \noindent We recall the irreducible finite dimensional modules for $U_{q}(sl_2)$. We use the following notation. \beast \lbrack n \rbrack_q = {{q^n - q^{-n}}\over {q-q^{-1}}} \qquad \qquad n \in \Z . \eeast \begin{lemma} \cite[p. 128]{Kassel} \label{lem:uqmods} With reference to Definition \ref{def:uqsl2}
there exists a family \begin{eqnarray} V_{\varepsilon,d} \qquad \quad \varepsilon \in \lbrace 1
-1\rbrace
\qquad \quad d = 0
2\ldots \label{eq:uqmods} \end{eqnarray} of irreducible finite dimensional $U_{q}(sl_2)$-modules with the following properties. The module $V_{\varepsilon,d}$ has a basis $u_0
u_1
u_d$ satisfying $ku_i=\varepsilon q^{d-2i}u_i$ for $0 \leq i \leq d$
$fu_i=\lbrack i+1\rbrack_q u_{i+1}$ for $0 \leq i \leq d-1$
$fu_d=0$
$eu_i=\varepsilon\lbrack d-i+1\rbrack_q u_{i-1}$ for $1 \leq i \leq d$
$eu_0=0$. Every irreducible finite dimensional $U_{q}(sl_2)$-module is isomorphic to exactly one of the modules $V_{\varepsilon,d}$. (Referring to line (\ref{eq:uqmods})
if $\K$ has characteristic 2 we interpret the set $\lbrace 1
-1 \rbrace $ as having a single element.) \end{lemma} %\medskip %\noindent %We use the above $U_q(sl_2)$-modules to get Leonard pairs. %The following result was proved by the present author in %\cite{Terint} %and is implicit in the results of Koelink and Van der Jeugt %\cite{cite38}
\cite{cite39}. For related results see %\cite{cite37}
%\cite{cite40}
%\cite{Rosengren}
%\cite{qSerre}. \begin{theorem} \cite{cite38}
\cite{cite39}
\cite{Terint} \label{ex:uqex} Referring to Definition \ref{def:uqsl2} and Lemma \ref{lem:uqmods}
let $\alpha
\beta $ denote nonzero scalars in $\K $ and define $A$
$A^*$ as follows. \begin{eqnarray*} A= \alpha f + {{ k}\over {q-q^{-1}}}
\qquad \qquad A^*=\beta e + {{k^{-1}}\over {q-q^{-1}}}. \label{eq:abdef} \end{eqnarray*} Let $d$ denote a nonnegative integer and choose $\varepsilon \in \lbrace 1
-1 \rbrace $. Then the pair $A
A^*$ acts on $V_{\varepsilon,d}$ as a Leonard pair provided $\varepsilon \alpha \beta $ is not among $q^{d-1}
q^{d-3}
q^{1-d}$. \end{theorem} \noindent We remark the Leonard pairs in Theorem \ref{ex:uqex} correspond to the quantum $q$-Krawtchouk polynomials \cite{KoeSwa}
\cite{cite37}. \section{Leonard pairs in combinatorics} Leonard pairs arise in many branches of combinatorics. For instance they arise in the theory of partially ordered sets (posets). We illustrate this with a poset called the subspace lattice $L_n(q)$. \medskip \noindent In this section we assume our field $\K$ is the field $\C$ of complex numbers. \medskip \noindent To define the subspace lattice we introduce a second field. Let $GF(q)$ denote a finite field of order $q$. Let $n$ denote a positive integer and let $W$ denote an $n$-dimensional vector space over $GF(q)$. Let $P$ denote the set consisting of all subspaces of $W$. The set $P$
together with the containment relation
is a poset called $L_n(q)$. \medskip \noindent Using $L_n(q)$ we obtain a family of Leonard pairs as follows. Let $\C P$ denote the vector space over $\C$ consisting of all formal $\C$-linear combinations of elements of $P$. We observe $P$ is a basis for $\C P$ so the dimension of $\C P$ is equal to the cardinality of P. \medskip \noindent We define three linear transformations on $\C P$. We call these $K$
$R$ (for ``raising'')
$L$ (for ``lowering''). \medskip \noindent We begin with $K$. For all $x \in P$
\beast K x = q^{{n/2} - dim\, x} x. % \qquad \qquad (x \in P). \eeast Apparently each element of $P$ is an eigenvector for $K$. \medskip \noindent To define $R$ and $L$ we use the following notation. For $x
y \in P$ we say $y$ {covers} $x$ whenever (i) $x \subseteq y$ and (ii) $\hbox{dim}\
y= 1+\hbox{dim}\
x$. \medskip \noindent The maps $R$ and $L$ are defined as follows. For all $x \in P$
\beast % R x = \sum_{{y \in P}\atop{y \;covers\; x}} y. R x = \sum_{y \;covers \; x} y. \eeast %where the sum is over all elements y of P that cover x. Similarly %for all $x \in P$
\beast % L x =q^{(1-n)/2} \sum_{{y \in P}\atop{x \;\hbox{covers}\; y}} y. L x =q^{(1-n)/2} \sum_{x \; covers \; y} y. \eeast %where the sum is over all elements y of P that are covered by x. (The scalar $q^{(1-n)/2}$ is included for aesthetic reasons.) \medskip \noindent We consider the properties of $K
R
L$. From the construction we find $K^{-1}$ exists. By combinatorial counting we verify \beast && K L = q L K
\qquad \qquad \qquad K R = q^{-1} R K
&& \qquad L R - R L = \frac{K - K^{-1}}{q^{1/2} - q^{-1/2}}. \eeast We recognize these equations. They are the defining relations for $U_{q^{1/2}}(sl_2)$. Apparently $K$
$R$
$L$ turn $\C P$ into a module for $U_{q^{1/2}}(sl_2)$. \medskip \noindent We now see how to get Leonard pairs from $L_n(q)$. Let $\alpha
\beta$ denote nonzero complex scalars and define $A$
$A^*$ as follows. \beast A = \alpha R + \frac{K}{q^{1/2}-q^{-1/2}}
\qquad \qquad A^* = \beta L + \frac{K^{-1}}{q^{1/2}-q^{-1/2}}. \eeast To avoid degenerate situations we assume $\alpha \beta $ is not among $q^{(n-1)/2}
q^{(n-3)/2}
q^{(1-n)/2}$. \medskip \noindent The $U_{q^{1/2}}(sl_2)$-module $\C P$ is completely reducible \cite[p. 144]{Kassel}. In other words $\C P$ is a direct sum of irreducible $U_{q^{1/2}}(sl_2)$-modules. On each irreducible module in this sum the pair $A
A^*$ acts as a Leonard pair. This follows from Theorem \ref{ex:uqex}. \medskip \noindent We just saw how the subspace lattice gives Leonard pairs. It is implicit in \cite{uniform} that the following posets give Leonard pairs in a similar fashion: the subset lattice
the Hamming semi-lattice
the attenuated spaces
and the classical polar spaces. Definitions of these posets can be found in \cite{uniform}. \section{Further reading} We mention some additional topics which are related to Leonard pairs. \medskip \noindent Earlier in this paper we obtained Leonard pairs from the irreducible finite dimensional modules for the Lie algebra $sl_2$ and the quantum algebra $U_q(sl_2)$. We cite some other algebras whose modules are related to Leonard pairs. These are the Askey-Wilson algebra \cite{GYZnature}
\cite{GYLZmut}
\cite{GYZTwisted}
\cite{GYZlinear}
\cite{GYZspherical}
\cite{Zhidd}
\cite{ZheCart}
\cite{Zhidden}
the Onsager algebra \cite{CKOnsn}
\cite{DateRoan2}
\cite{Dav}
\cite{Dolgra}
and the Tridiagonal algebra \cite{TD00}
\cite{LS99}
\cite{qSerre}. \medskip \noindent We discussed how certain classical posets give Leonard pairs. Another combinatorial object which gives Leonard pairs is a $P$- and $Q$-polynomial association scheme \cite{BanIto}
\cite{bcn}
\cite{TersubI}. Leonard pairs have been used to describe certain irreducible modules for the subconstituent algebra of these schemes \cite{Cau}
\cite{CurNom}
\cite{Curspin}
\cite{TD00}
\cite{TersubI}. \medskip \noindent The topic of Leonard pairs is closely related to the work of Gr\"unbaum and Haine on the ``bispectral problem'' \cite{GH7}
\cite{GH6}. See \cite{GH4}
\cite{GH5}
\cite{GH1}
\cite{GH3}
\cite{GH2} for related work. {\begin{thebibliography}{10} \bibitem{CKOnsn} C.~Ahn and K.~Shigemoto
\newblock Onsager algebra and integrable lattice models
\newblock {\em Modern Phys. Lett. A} {6(38)} (1991) 3509--3515. \bibitem{Ask} G.~Andrews
R.~Askey
and R.~Roy
\newblock {\em Special functions}
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1999. \bibitem{BanIto} E.~Bannai and T.~Ito
\newblock {\em Algebraic Combinatorics {I}: Association Schemes}
\newblock Benjamin/Cummings
London
1984. \bibitem{bcn} A.~E. Brouwer
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and A.~Neumaier
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\newblock Springer-Verlag
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\newblock Distance-regular graphs which support a spin model are thin
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\newblock The structure of quotients of the {O}nsager algebra by closed ideals
\newblock {\em J. Phys. A: Math. Gen.} {33} (2000) 3275--3296. \bibitem{Dav} B.~Davies
\newblock Onsager's algebra and the {D}olan-{G}rady condition in the non-self-dual case
\newblock {\em J. Math. Phys.} {32} (1991) 2945--2950. \bibitem{Dolgra} L.~Dolan and M.~Grady
\newblock Conserved charges from self-duality
\newblock {\em Phys. Rev. D (3)} {25} (1982) 1587--1604. \bibitem{GR} G.~Gasper and M.~Rahman
\newblock {\em Basic hypergeometric series}. \newblock Encyclopedia of Mathematics and its Applications
35
1990. \bibitem{GYZnature} Ya.~A. Granovski{\u\i} and A.~S. Zhedanov
\newblock Nature of the symmetry group of the $6j$-symbol
\newblock {\em Zh. \`Eksper. Teoret. Fiz.} {94} (1988) 49--54. \bibitem{GYLZmut} Ya.~I. Granovski{\u\i}
I.~M. Lutzenko
and A.~S. Zhedanov
\newblock Mutual integrability
quadratic algebras
and dynamical symmetry
\newblock {\em Ann. Physics} {217} (1992) 1--20. \bibitem{GYZTwisted} Ya.~I. Granovski{\u\i} and A.~S. Zhedanov
\newblock ``{T}wisted'' {C}lebsch-{G}ordan coefficients for ${{s}{u}}\sb q(2)$
\newblock {\em J. Phys. A} {25} (1992) L1029--L1032. \bibitem{GYZlinear} Ya.~I. Granovski{\u\i} and A.~S. Zhedanov
\newblock Linear covariance algebra for ${{s}{l}}\sb q(2)$
\newblock {\em J. Phys. A} {26} (1993) L357--L359. \bibitem{GYZspherical} Ya.~I. Granovski{\u\i} and A.~S. Zhedanov
\newblock Spherical $q$-functions
\newblock {\em J. Phys. A} {26} (1993) 4331--4338. \bibitem{GH4} F.~A.~ Gr{\"u}nbaum
\newblock Some bispectral musings
in: \newblock {\em The bispectral problem (Montreal, PQ, 1997)}
Amer. Math. Soc.
Providence
RI
1998
pp. 31--45. \bibitem{GH5} F.~A.~Gr{\"u}nbaum and L.~Haine
\newblock Bispectral {D}arboux transformations: an extension of the {K}rall polynomials
\newblock {\em Internat. Math. Res. Notices} {8} (1997) 359--392. \bibitem{GH7} F.~A.~Gr{\"u}nbaum and L.~Haine
\newblock The $q$-version of a theorem of {B}ochner
\newblock {\em J. Comput. Appl. Math.} {68} (1996) 103--114. \bibitem{GH6} F.~A.~Gr{\"u}nbaum and L.~Haine
\newblock Some functions that generalize the {A}skey-{W}ilson polynomials
\newblock {\em Comm. Math. Phys.} {184} (1997) 173--202. \bibitem{GH1} F.~A.~Gr{\"u}nbaum and L.~Haine
\newblock On a $q$-analogue of the string equation and a generalization of the classical orthogonal polynomials
in: \newblock {\em Algebraic methods and $q$-special functions (Montr\'eal, QC, 1996)} Amer. Math. Soc.
1999
pp. 171--181. \bibitem{GH3} F.~A.~Gr{\"u}nbaum and L.~Haine
\newblock The {W}ilson bispectral involution: some elementary examples
in: \newblock {\em Symmetries and integrability of difference equations (Canterbury, 1996)}
Cambridge Univ. Press
pp. 353--369. \bibitem{GH2} F.~A.~Gr{\"u}nbaum
L.~Haine
and E.~Horozov
\newblock Some functions that generalize the {K}rall-{L}aguerre polynomials
\newblock {\em J. Comput. Appl. Math.} {106} (1999) 271--297. \bibitem{TD00} T.~Ito
K.~Tanabe
and P.~Terwilliger
\newblock Some algebra related to ${P}$- and ${Q}$-polynomial association schemes
in: \newblock {\em Codes and Association Schemes (Piscataway NJ, 1999)}
Providence RI
2001
pp. 167--192. \bibitem{shape} T.~Ito and P.~Terwilliger. \newblock The shape of a tridiagonal pair. \newblock {\em J. Pure Appl. Algebra}
submitted. Available at \newblock{ http://front.math.ucdavis.edu/math.QA/0304244 } \bibitem{Kassel} C.~Kassel
\newblock {\em Quantum groups}
New York
1995. \bibitem{KoeSwa} R.~Koekoek and R.~F.~Swarttouw
\newblock {\em The Askey scheme of hypergeometric orthogonal polyomials and its $q$-analog}
report 98-17
The Netherlands
1998. Available at \newblock{ http://aw.twi.tudelft.nl/{\~{}}koekoek/research.html} \bibitem{Koelink3} H.~T. Koelink
\newblock Askey-{W}ilson polynomials and the quantum ${{s}{u}}(2)$ group: survey and applications
\newblock {\em Acta Appl. Math.} {44} (1996) 295--352. \bibitem{cite37} H.~T. Koelink
\newblock $q$-Krawtchouk polynomials as spherical functions on the Hecke algebra of type $B$
\newblock {\em Trans. Amer. Math. Soc.} {352} (2000) 4789--4813. \bibitem{cite38} H.~T. Koelink and J.~Van der Jeugt
\newblock Convolutions for orthogonal polynomials from Lie and quantum algebra representations
\newblock {\em SIAM J. Math. Anal.} {29} (1998) 794--822. \bibitem{cite39} H.~T. Koelink and J.~Van der Jeugt
\newblock Bilinear generating functions for orthogonal polynomials
\newblock {\em Constr. Approx.} {15} (1999) 481--497. \bibitem{cite40} T.~H.~Koornwinder
\newblock Askey-Wilson polynomials as zonal spherical functions on the su(2) quantum group
\newblock {\em SIAM J. Math. Anal.} {24} (1993) 795--813. \bibitem{Leon} D.~Leonard
\newblock Orthogonal polynomials
duality
and association schemes
\newblock {\em SIAM J. Math. Anal.} {13} (1982) 656--663. \bibitem{Rosengren} H.~Rosengren
\newblock {\em Multivariable orthogonal polynomials as coupling coefficients for Lie and quantum algebra representations.} \newblock Centre for Mathematical Sciences
Sweden
1999. \bibitem{uniform} P.~Terwilliger
\newblock The incidence algebra of a uniform poset
\newblock {\em Math and its applications} {20} (1990) 193--212. \bibitem{TersubI} P.~Terwilliger
\newblock The subconstituent algebra of an association scheme
\newblock {\em J. Algebraic Combin.} {1} (1992) 363--388. \bibitem{LS99} P.~Terwilliger. \newblock Two linear transformations each tridiagonal with respect to an eigenbasis of the other. \newblock {\em Linear Algebra Appl.} {330} (2001)
149--203. \bibitem{qSerre} P.~Terwilliger. \newblock Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations. In \newblock {\em Physics and Combinatorics 1999 (Nagoya)}
377--398
World Scientific Publishing
River Edge
NJ
2001 \bibitem{LS24} P.~Terwilliger. \newblock Leonard pairs from 24 points of view. \newblock {\em Rocky Mountain J. Math.} {32}(2) (2002)
827--888. \bibitem{conform} P.~Terwilliger. \newblock Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the $TD$-$D$ and the $LB$-$UB$ canonical form. \newblock {\em J. Algebra}. Submitted. Available at \newblock{ http://front.math.ucdavis.edu/math.RA/0304077 } \bibitem{lsint} P.~Terwilliger. \newblock Introduction to Leonard pairs. \newblock {OPSFA Rome 2001}. \newblock{\em J. Comput. Appl. Math.} {153}(2) (2003)
463--475. \bibitem{Terint} P.~Terwilliger. \newblock Introduction to {L}eonard pairs and {L}eonard systems. \newblock {\em S\=urikaisekikenky\=usho K\=oky\=uroku}
\newblock (1109):67--79
1999. \newblock Algebraic combinatorics (Kyoto
1999). \bibitem{TLT:split} P.~Terwilliger. \newblock Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the split decomposition. \newblock {\em Indag. Math.} Submitted. Available at \newblock{ http://front.math.ucdavis.edu/math.RA/0306290 } \bibitem{TLT:array} P.~Terwilliger. \newblock Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array. \newblock {\em Geometric and Algebraic Combinatorics 2, Oisterwijk, The Netherlands 2002}. \newblock Submitted. Available at \newblock{ http://front.math.ucdavis.edu/math.RA/0306291 } \bibitem{qrac} P.~Terwilliger. \newblock Leonard pairs and the $q$-Racah polynomials. \newblock {\em Linear Algebra Appl.} Submitted. \bibitem{aw} P.~Terwilliger and R.~Vidunas. \newblock Leonard pairs and the Askey-Wilson relations. \newblock {\em J. Algebra Appl.} Submitted. Available at \newblock{ http://front.math.ucdavis.edu/math.QA/0305356 } \bibitem{Zhidd} A.~S. Zhedanov
\newblock ``{H}idden symmetry'' of {A}skey-{W}ilson polynomials
\newblock {\em Teoret. Mat. Fiz.} {89} (1991) 190--204. \bibitem{ZheCart} A.~S. Zhedanov
\newblock Quantum ${{s}{u}}\sb q(2)$ algebra: ``{C}artesian'' version and overlaps
\newblock {\em Modern Phys. Lett. A} {7} (1992) 1589--1593. \bibitem{Zhidden} A.~S. Zhedanov
\newblock Hidden symmetry algebra and overlap coefficients for two ring-shaped potentials
\newblock {\em J. Phys. A} {26} (1993) 4633--4641. \end{thebibliography} } \noindent Paul Terwilliger
Department of Mathematics
480 Lincoln Drive
Madison
Wisconsin
53706
Let $\K$ denote a field and let $V$ denote a vector space over $\K$ with finite positive dimension. We consider an ordered pair of linear transformations $A:V\rightarrow V$ and $A^*:V\rightarrow V$ that satisfy conditions (i), (ii) below. \begin{enumerate} em There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal. em There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. \end{enumerate} We call such a pair a {Leonard pair} on $V$. We give an overview of the theory of Leonard pairs.
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%{\Large {
\title{ \bf Two linear transformations each tridiagonal \\ 
 \bf with respect to an eigenbasis of the other;\\
 an overview 
%\title{ \bf Introduction to Leonard pairs\footnote{
%{\bf Keywords}. Leonard pair, Tridiagonal pair, Askey scheme, Askey-Wilson
%polynomial, $q$-Racah polynomial.
%\hfil\break
%\noindent 
%{\bf 2000 Mathematics Subject Classification}. 
%05E30, 05E35, 17B37, 33C45, 33D45.
}
%\author{Paul Terwilliger\footnote{  
\author{Paul Terwilliger}
%\footnote{  
% Mathematics  Department,  
%University of  Wisconsin, 
% 480 Lincoln Drive, 
%Madison, WI 53706 \ \
%Email:terwilli@math.wisc.edu }
%}
\date{}
%to get date printout, comment out above line 
\maketitle
\begin{abstract} 
Let $\K$ denote a field 
 and let $V$ denote a  
vector space over $\K$ with finite positive dimension.
We consider an ordered pair of linear transformations 
$A:V\rightarrow V$ and $A^*:V\rightarrow V$ 
that 
 satisfy conditions (i), (ii) below. 
\begin{enumerate}
\item There exists a basis for $V$ with respect to which
the matrix representing $A$ is irreducible tridiagonal and
the matrix representing $A^*$ is diagonal.
\item There exists a basis for $V$ with respect to which
the matrix 
representing $A$ is diagonal and 
the matrix representing $A^*$ is irreducible tridiagonal.
\end{enumerate}
We call such a pair a {\it Leonard pair} on $V$.
We give an overview of the theory of Leonard pairs.
\end{abstract}


\section{Leonard pairs}
\medskip
%\noindent Throughout  $\K$ will denote an arbitrary
%field.

\medskip
\noindent We begin by recalling the notion of a Leonard pair.
We will use the following terms. Let $X$ denote a square matrix.
Then $X$ is called {\it tridiagonal} whenever each nonzero
entry lies on either the diagonal, the subdiagonal, or the
superdiagonal. Assume $X$ is tridiagonal. Then $X$ is
called {\it irreducible} whenever each entry on
the subdiagonal is nonzero and each entry on the superdiagonal
is nonzero.

\medskip
\noindent We now define a Leonard pair.  For the rest of this paper
$\K$ will denote a field.

\begin{definition} 
\label{def:lprecall}
\cite{LS99}
\rm
Let  
 $V$ denote a  
vector space over $\K$ with finite positive dimension.
By a {\it Leonard pair} on $V$
we mean an ordered pair of linear transformations 
$A:V\rightarrow V$ and $A^*:V\rightarrow V$ 
that 
 satisfy conditions (i), (ii) below. 
\begin{enumerate}
\item There exists a basis for $V$ with respect to which
the matrix representing $A$ is irreducible tridiagonal and
the matrix representing $A^*$ is diagonal.
\item There exists a basis for $V$ with respect to which
the matrix 
representing $A$ is diagonal and 
the matrix representing $A^*$ is irreducible tridiagonal.
\end{enumerate}
\end{definition}

\begin{note}
\rm
According to a common notational convention,
$A^*$ denotes the conjugate transpose of $A$.
We are not using this convention.
In a Leonard pair $A,A^*$ the linear transformations $A,A^*$
are arbitrary subject to (i), (ii) above.
\end{note}

\noindent 
\begin{note}
\rm
Our use of the name ``Leonard pair'' is motivated by a connection
to a theorem of D. Leonard 
\cite[p.  260]{BanIto}, \cite{Leon}
involving the $q$-Racah and related
polynomials of the Askey Scheme.
%See 
%\cite[App. A]{LS99} and 
%\cite{LS24} for more information
%on this.
\end{note}

%\begin{note} According to a common notational convention,
%for a linear transformation $A$ the conjugate-transpose of $A$
%is denoted $A^*$. We emphasize we are not using this convention.
%In a Leonard pair $A,A^*$, the linear transformations $A,A*$
%are arbitrary subject to (i), (ii) above.
%\end{note}

\section{An example of a Leonard pair} 
\medskip
\noindent
Here is an example of a Leonard pair.
Set 
$V={\K}^4$ (column vectors), set 
\beast
A = 
\left(
\begin{array}{ c c c c }
0 & 3  &  0    & 0  \\
1 & 0  &  2   &  0    \\
0  & 2  & 0   & 1 \\
0  & 0  & 3  & 0 \\
\end{array}
\right), \qquad  
A^* = 
\left(
\begin{array}{ c c c c }
3 & 0  &  0    & 0  \\
0 & 1  &  0   &  0    \\
0  & 0  & -1   & 0 \\
0  & 0  & 0  & -3 \\
\end{array}
\right),
\eeast
and view $A$ and $A^*$  as linear transformations on $V$.
We assume 
the characteristic of $\K$ is not 2 or 3 to ensure
$A$ is irreducible.
Then the pair $A, A^*$ is a Leonard
pair on $V$. 
Indeed 
condition (i) in Definition
\ref{def:lprecall}
is satisfied by the basis for $V$
consisting of the columns of the 4 by 4 identity matrix.
To verify condition (ii), we display an invertible  matrix  
$P$ such that 
$P^{-1}AP$ is 
diagonal and 
$P^{-1}A^*P$ is
irreducible tridiagonal.
Set 
\beast
P = 
\left(
\begin{array}{ c c c c}
1 & 3  &  3    &  1 \\
1 & 1  &  -1    &  -1\\
1  & -1  & -1  & 1  \\
1  & -3  & 3  & -1 \\
\end{array}
\right).
\eeast
 By matrix multiplication $P^2=8I$, where $I$ denotes the identity,   
so $P^{-1}$ exists. Also by matrix multiplication,    
\begin{equation}
AP = PA^*.
\label{eq:apeq}
\end{equation}
Apparently
$P^{-1}AP$ is equal to $A^*$ and is therefore diagonal.
By (\ref{eq:apeq}) and since $P^{-1}$ is
a scalar multiple of $P$, we find
$P^{-1}A^*P$ is equal to $A$ and is therefore irreducible tridiagonal.  Now 
condition (ii) of  Definition 
\ref{def:lprecall}
is satisfied
by the basis for $V$ consisting of the columns of $P$. 

\section{Leonard systems}

When working with a Leonard pair, it is often convenient to
consider a closely related and somewhat more abstract concept
called a {\it Leonard system}.
In order to define this we recall a few terms.
Let $V$ denote a vector space over $\K$ with finite positive
dimension. Let $\mbox{End}(V)$ denote the $\K$-algebra
consisting of the linear transformations from $V$ to $V$.
For $A \in 
 \mbox{End}(V)$, by the {\it eigenvalues} of $A$ we mean 
 the roots of the characteristic polynomial of $A$. These eigenvalues
 are contained in the algebraic closure of $\K$.
 We say $A$ is {\it multiplicity-free}
 whenever the eigenvalues of $A$ are mutually distinct and contained in
 $\K$. Assume for the moment that $A$ is multiplicity-free.
Let $\theta_0,\theta_1, \ldots, \theta_d$ denote an ordering
of the eigenvalues of $A$. For $0 \leq i \leq d$ let 
$v_i$ denote a nonzero vector in  $V$ that is an eigenvector
for $A$ with eigenvalue $\theta_i$.
Observe the sequence $v_0, v_1, \ldots, v_d$ is a basis for $V$.
For $0 \leq i \leq d$ define 
$E_i \in \mbox{End}(V)$
so that $E_iv_j=\delta_{ij}v_j$ for $0 \leq j\leq d$.
We call $E_i$ the {\it primitive idempotent} of $A$ associated with
$\theta_i$.
\begin{definition}
\label{def:ls}
\cite{LS99}
\rm
Let $d$ denote a nonnegative integer and 
let $V$ denote a vector space over $\K$ with dimension $d+1$.
By a {\it Leonard system} on $V$ we mean a sequence
$\Phi=(A;A^*; $ $\lbrace E_i \rbrace_{i=0}^d; 
\lbrace E^*_i \rbrace_{i=0}^d)$ that satisfies 
conditions (i)--(v) below.
\begin{enumerate}
\item Each of $A,A^*$ is a multiplicity-free element of
 $\mbox{End}(V)$.
\item $E_0, E_1, \ldots, E_d$ is an ordering of the primitive idempotents
of $A$.
\item $E^*_0, E^*_1, \ldots, E^*_d$ is an ordering of the primitive idempotents
of $A^*$.
\item 
${\displaystyle{
E^*_iAE^*_j = \cases{0, &if $\;|i-j|> 1$;\cr
\not=0, &if $\;|i-j|=1$\cr}
\qquad \qquad 
(0 \leq i,j\leq d).
}}$
\item
${\displaystyle{
 E_iA^*E_j = \cases{0, &if $\;|i-j|> 1$;\cr
\not=0, &if $\;|i-j|=1$\cr}
\qquad \qquad 
(0 \leq i,j\leq d).
}}$
\end{enumerate}
We call $d$ the {\it diameter} of $\Phi$. 
\end{definition}
Leonard pairs and Leonard systems are related as follows.

\begin{theorem}
\cite{TLT:split}
For $A,A^*$ in 
 $\mbox{End}(V)$,
 the pair $A,A^*$
is a Leonard pair on $V$ if and only if
the following (i), (ii) hold.
\begin{enumerate}
\item Each of $A,A^*$ is multiplicity-free.
\item There exists an ordering $E_0, E_1, \ldots, E_d$
of the primitive idempotents of $A$
and there exists an ordering
 $E^*_0, E^*_1, \ldots, E^*_d$
of the primitive idempotents of $A^*$ such that
$(A;A^*;$ $\lbrace E_i \rbrace_{i=0}^d; 
\lbrace E^*_i \rbrace_{i=0}^d)$ is a Leonard system on $V$.
\end{enumerate}
\end{theorem}

\section{Leonard pairs from 24 points of view}

\medskip
Let $A, A^*$ denote a Leonard 
pair on $V$.
We describe  
24 bases for $V$ on which $A,A^*$ act in an attractive fashion.

\medskip
\noindent 
To describe the bases
we will use the following terms.
Let $X$ denote a square matrix. Then $X$ is called
{\it lower bidiagonal} 
 whenever each nonzero
entry lies on either the diagonal or the subdiagonal.
We say $X$ is 
{\it upper bidiagonal} 
 whenever the transpose of $X$ is lower bidiagonal.
Let $v_0, v_1, \ldots, v_d$ denote
a  basis for $V$.
By 
the {\it inversion} of 
this basis
we mean the basis $v_d, v_{d-1}, \ldots, v_0$.

\medskip
\noindent The 24 bases are described by the diagram below.
In that diagram each  vertex represents one  of the
24 bases.
The shading on the vertex  indicates the nature
of the
$A,A^*$ action.
\begin{enumerate}
\item
Black: 
 $A$ is 
 diagonal
and  $A^*$
is irreducible tridiagonal.
\item Green: 
$A$ is lower bidiagonal and $A^*$ is upper bidiagonal.
\item
Red: 
$A$ is upper bidiagonal and $A^*$ is lower bidiagonal.
\item Yellow:
 $A$ is irreducible tridiagonal 
and $A^*$ is diagonal.
\end{enumerate}

\noindent
For each pair  
 of bases in the 
diagram that are connected by an arc,
consider the transition matrix from
one of these bases to the other. 
The shading on the arc indicates the nature of
this transition matrix. 
\begin{enumerate}
\item Solid arc: Transition matrix
is diagonal. 
\item
Dashed arc:
Transition matrix
is lower triangular.
\item
Dotted arc: The bases are the inversion
of one another.
\end{enumerate}

\bigskip
\input psfig.sty
\centerline{\psfig{figure=s4oct1.eps,height=4.5cm}} 
\vskip .5cm

\bigskip
\noindent See
   \cite{LS24} for more information concerning the 24 bases.



\section{The classifying space}

We will shortly give a classification of the Leonard systems.
In order to describe the result we recall the notion
of a {\it parameter array}.

\begin{definition} 
%\cite[Definition 10.1]{conform}
\label{def:pa}
\rm
Let $d$ denote a nonnegative integer.
By a {\it parameter array over $\K$ of diameter $d$}
we mean a sequence of scalars 
$
(\theta_i, \theta^*_i,i=0..d; \varphi_j, \phi_j,j=1..d)
$
taken from $\K$ which satisfy the following conditions
(PA1)--(PA5).
\begin{description}
\item[(PA1)]  
$
\theta_i \not=\theta_j, \qquad 
\theta^*_i \not=\theta^*_j \qquad \mbox{if}\quad i\not=j, \qquad \qquad
(0 \leq i,j\leq d)
$.
\\
\item[(PA2)]
$
\varphi_i \not=0, \qquad \phi_i \not=0 \qquad \qquad (1 \leq i \leq d).
$
\\
\item[(PA3)]
$\varphi_i = \phi_1 \sum_{h=0}^{i-1} 
\frac{\theta_h-\theta_{d-h}}{\theta_0-\theta_d}
+ (\theta^*_i-\theta^*_0)(\theta_{i-1}-\theta_d)
\qquad \qquad (1 \leq i \leq d)$.
\\
\item[(PA4)] 
$\phi_i = \varphi_1 \sum_{h=0}^{i-1} 
\frac{\theta_h-\theta_{d-h}}{\theta_0-\theta_d}
+ (\theta^*_i-\theta^*_0)(\theta_{d-i+1}-\theta_0)
\qquad \qquad (1 \leq i \leq d)$.
\\
\item[(PA5)]
The expressions
\begin{eqnarray*} 
\frac{\theta_{i-2}-\theta_{i+1}}{\theta_{i-1}-\theta_i},
\qquad 
\frac{\theta^*_{i-2}-\theta^*_{i+1}}{\theta^*_{i-1}-\theta^*_i}
\label{eq:betaplusone}
\end{eqnarray*}
are equal and independent of $i$ for $2 \leq i \leq d-1$.
\end{description}
\end{definition}




\section{The classification of Leonard systems}

In this section we give a bijection from
the set of parameter arrays to the set of isomorphism
classes of Leonard systems.
Let
$(\theta_i, \theta^*_i,i=0..d;\varphi_j,\phi_j,j=1..d)$
denote a parameter array over $\K$.
Define
\begin{eqnarray*}
A=\left(
\begin{array}{c c c c c c}
\theta_0 & & & & & {\mathbf 0} \\
1 & \theta_{1} &  & & & \\
& 1 & \theta_{2} &  & & \\
& & \cdot & \cdot &  &  \\
& & & \cdot & \cdot &  \\
{\mathbf 0}& & & & 1 & \theta_d
\end{array}
\right),
\quad A^*=\left(
\begin{array}{c c c c c c}
\theta^*_0 &\varphi_1 & & & & {\mathbf 0} \\
 & \theta^*_1 & \varphi_2 & & & \\
&  & \theta^*_2 & \cdot & & \\
& &  & \cdot & \cdot &  \\
& & &  & \cdot & \varphi_d \\
{\mathbf 0}& & & &  & \theta^*_d
\end{array}
\right).
\end{eqnarray*}
Observe 
$A$ (resp. $A^*$) is multiplicity free
with eigenvalues
$\theta_0, \theta_1, \ldots, \theta_d$
(resp.
$\theta^*_0, \theta^*_1, \ldots, \theta^*_d$).
For $0 \leq i \leq d$ let $E_i$ (resp. $E^*_i$)
denote the primitive idempotent of $A$ (resp. $A^*$)
associated with $\theta_i$ (resp. $\theta^*_i$).
Then the sequence
$(A;A^*; $ $\lbrace E_i \rbrace_{i=0}^d; 
\lbrace E^*_i \rbrace_{i=0}^d)$ 
is a Leonard system on $\K^{d+1}$.
This construction induces a bijection from
the set of parameter arrays to the set of isomorphism
classes of Leonard systems \cite{LS99}.



\section{Leonard pairs $A,A^*$ with $A$ lower bidiagonal and
$A^*$ upper bidiagonal}

\noindent 
Let $A, A^*$ denote
matrices in $\hbox{Mat}_{d+1}(\K)$. Let us assume 
$A$ is lower bidiagonal 
and $A^*$ is upper  bidiagonal. 
We give a necessary and sufficient condition
for $A,A^*$ to be  a Leonard pair. 


\begin{theorem}
   \cite{conform}
\label{thm:lug}
Let $d$ denote a nonnegative integer and let $A, A^*$ denote
matrices in $\hbox{Mat}_{d+1}(\K)$. Assume $A$  
lower bidiagonal and $A^*$ is upper bidiagonal.
Then the following (i), (ii) are equivalent.
\begin{enumerate}
\item
The pair $A,A^*$ is a Leonard pair.
\item
There exists a parameter array
$(\theta_i, \theta^*_i, i=0..d;  \varphi_j, \phi_j, j=1..d)$
over $\K$ 
such that
\begin{eqnarray*}
&&A_{ii} =\theta_i,
\qquad \qquad 
A^*_{ii} =\theta^*_i \qquad \qquad (0 \leq i \leq d),
\label{eq:comb1}
\\
&&\qquad A_{i,i-1}A^*_{i-1,i} = \varphi_i \qquad \qquad (1 \leq i \leq d).
\label{eq:comb2}
\end{eqnarray*}
\end{enumerate}
\end{theorem}

\section{Leonard pairs $A,A^*$ with $A$ tridiagonal and $A^*$ diagonal}

\noindent 
Let $A, A^*$ denote
matrices in $\hbox{Mat}_{d+1}(\K)$. Let us assume 
$A$ is tridiagonal 
and $A^*$ is diagonal. 
We give  a necessary and sufficient condition for
 $A,A^*$ to be a Leonard pair. 

\begin{theorem}
   \cite{conform}
\label{thm:tdcrit}
Let $d$ denote a nonnegative integer and let $A, A^*$ denote
matrices in $\hbox{Mat}_{d+1}(\K)$. Assume $A$ is
tridiagonal and $A^*$ is diagonal.
Then the following (i), (ii) are equivalent.
\begin{enumerate}
\item
The pair $A,A^*$ is a Leonard pair.
\item
There exists
a parameter array 
$(\theta_i, \theta^*_i, i=0..d;  \varphi_j, \phi_j, j=1..d)$
over $\K$
such that
\begin{eqnarray*}
A_{ii} &=&
\theta_i + \frac{\varphi_i}{\theta^*_i-\theta^*_{i-1}}
+
 \frac{\varphi_{i+1}}{\theta^*_i-\theta^*_{i+1}}
\qquad \qquad  
(0 \leq i \leq d), 
\label{eq:aiiform}
\\
A_{i,i-1}A_{i-1,i}&=&
\varphi_i\phi_i \frac{\prod_{h=0}^{i-2}(\theta^*_{i-1}-\theta^*_h)
}
{\prod_{h=0}^{i-1}(\theta^*_{i}-\theta^*_h)
}
\, \frac{\prod_{h=i+1}^d (\theta^*_i-\theta^*_h)
}
{\prod_{h=i}^d(\theta^*_{i-1}-\theta^*_h)
}
\qquad  (1 \leq i \leq d),
\label{eq:crossprod}
\\
A^*_{ii} &=&\theta^*_i \qquad \qquad (0 \leq i \leq d).
\label{eq:ths}
\end{eqnarray*}
\end{enumerate}
\end{theorem}



\section{A characterization of the parameter arrays I}

In this section we characterize the parameter
arrays in terms of bidiagonal matrices.
We will refer to the following set-up.

\begin{definition}
\label{def:setup}
Let $d$ denote a nonnegative integer and let
$(\theta_i, \theta^*_i,i=0..d; \varphi_j, \phi_j,j=1..d)
$
denote a sequence of  scalars taken from $\K$. We assume
this sequence satisfies 
PA1 and PA2.
\end{definition}


\begin{theorem}
 \cite{TLT:array}
\label{thm:gmat}
With reference to Definition \ref{def:setup},
 the following (i), (ii) are equivalent.
\begin{enumerate}
\item
The sequence
$(\theta_i, \theta^*_i,i=0..d;\varphi_j,\phi_j,j=1..d)$
satisfies PA3--PA5.
\item
There exists an invertible matrix 
$G \in \mbox{Mat}_{d+1}(\K)$ such that both
\begin{eqnarray*}
G^{-1}
\left(
\begin{array}{c c c c c c}
\theta_0 & & & & & {\mathbf 0} \\
1 & \theta_{1} &  & & & \\
& 1 & \theta_{2} &  & & \\
& & \cdot & \cdot &  &  \\
& & & \cdot & \cdot &  \\
{\mathbf 0}& & & & 1 & \theta_d
\end{array}
\right) G
&=&\left(
\begin{array}{c c c c c c}
\theta_d & & & & & {\mathbf 0} \\
1 & \theta_{d-1} &  & & & \\
& 1 & \theta_{d-2} &  & & \\
& & \cdot & \cdot &  &  \\
& & & \cdot & \cdot &  \\
{\mathbf 0}& & & & 1 & \theta_0
\end{array}
\right),
\label{eq:g1}
\\
G^{-1}
\left(
\begin{array}{c c c c c c}
\theta^*_0 &\varphi_1 & & & & {\mathbf 0} \\
 & \theta^*_1 & \varphi_2 & & & \\
&  & \theta^*_2 & \cdot & & \\
& &  & \cdot & \cdot &  \\
& & &  & \cdot & \varphi_d \\
{\mathbf 0}& & & &  & \theta^*_d
\end{array}
\right) G
&=&
\left(
\begin{array}{c c c c c c}
\theta^*_0 &\phi_1 & & & & {\mathbf 0} \\
 & \theta^*_1 & \phi_2 & & & \\
&  & \theta^*_2 & \cdot & & \\
& &  & \cdot & \cdot &  \\
& & &  & \cdot & \phi_d \\
{\mathbf 0}& & & &  & \theta^*_d
\end{array}
\right).
\label{eq:g2}
\end{eqnarray*}
\end{enumerate}
\end{theorem}

\section{A characterization of the parameter arrays II}

\noindent In this section we characterize the parameter
arrays in terms of polynomials.
We will use the following notation.
Let $\lambda $ denote an indeterminate, and let
$\K \lbrack \lambda \rbrack $ denote the $\K$-algebra
consisting of all polynomials in $\lambda $  which have
coefficients in $\K$. From now on
all polynomials which we discuss
are assumed to lie in $\K\lbrack \lambda \rbrack $. 


\begin{theorem}
 \cite{TLT:array}
\label{eq:mth}
With reference to Definition \ref{def:setup},
the following (i), (ii) are equivalent.
\begin{enumerate}
\item
The sequence
 $(\theta_i, \theta^*_i,i=0..d; \varphi_j, \phi_j,j=1..d)$
satisfies PA3--PA5.
\item For $0 \leq i \leq d$ the 
polynomial 
\begin{equation}
\sum_{n=0}^i \frac{
(\lambda-\theta_0)
(\lambda-\theta_1) \cdots
(\lambda-\theta_{n-1})
(\theta^*_i-\theta^*_0)
(\theta^*_i-\theta^*_1) \cdots
(\theta^*_i-\theta^*_{n-1})
}
{\varphi_1\varphi_2\cdots \varphi_n}
\label{eq:poly1}
\end{equation}
is a scalar multiple of the polynomial 
\begin{eqnarray*}
\sum_{n=0}^i \frac{
(\lambda-\theta_d)
(\lambda-\theta_{d-1}) \cdots
(\lambda-\theta_{d-n+1})
(\theta^*_i-\theta^*_0)
(\theta^*_i-\theta^*_1) \cdots
(\theta^*_i-\theta^*_{n-1})
}
{\phi_1\phi_2\cdots \phi_n}.
\label{eq:poly2}
\end{eqnarray*}
\end{enumerate}
\end{theorem}

\section{Some orthogonal polynomials of the Askey scheme}

There is a natural correspondence between 
Leonard systems and a class of orthogonal polynomials 
consisting of the $q$-Racah polynomials and some related
polynomials of the Askey scheme. This correspondence
is described as follows
 \cite{TLT:array}.

\medskip
\noindent
Let 
$\Phi$
%=(A;A^*; $ $\lbrace E_i \rbrace_{i=0}^d; 
%\lbrace E^*_i \rbrace_{i=0}^d)$ 
denote a Leonard system.  
By the {\it polynomials which correspond to $\Phi$} we mean 
the polynomials 
from  
(\ref{eq:poly1}), where the parameter array from
that line is the one associated with 
$\Phi$.
The polynomials which correspond to a Leonard system
are listed below:

\medskip
\noindent
$q$-Racah, $q$-Hahn, dual $q$-Hahn,
$q$-Krawtchouk,
dual $q$-Krawtchouk,
quantum $q$-Krawtchouk, 
affine $q$-Krawtchouk, 
Racah, Hahn, dual-Hahn, Krawtchouk, Bannai/Ito, and  orphan polynomials.

\medskip
\noindent
The Bannai/Ito polynomials can be obtained from the $q$-Racah polynomials by letting $q$ 
tend to $-1$. The orphan polynomials have maximal degree 3 and 
exist for $\mbox{char}(\K)=2$  only. 

\medskip
\noindent See 
 \cite{KoeSwa} for information on the Askey scheme.


\section{The Askey-Wilson relations}

We turn our attention to the representation theoretic aspects
of Leonard pairs.

\begin{theorem}
\cite{aw}
\label{lptheorem}
Let $V$ denote a vector space over $\K$ with finite positive
dimension. Let $A,A^*$ denote
a Leonard pair on $V$. Then there
exists a sequence of scalars
$\beta,\gamma,\gamma^*,\varrho,\varrho^*$, $\omega,\eta,\eta^*$
taken from $\K$ such that both
\begin{eqnarray*}  \label{askwil1}
A^2 A^*-\beta A A^*\!A+A^*\!A^2-\gamma\left( A A^*\!+\!A^*\!A
\right)-\varrho\,A^* &=& \gamma^*\!A^2+\omega A+\eta\,I,\\
\label{askwil2} A^*{}^2\!A-\beta A^*\!AA^*\!+AA^*{}^2-
\gamma^*\!\left(A^*\!A\!+\!A A^*\right)-\varrho^*\!A &=&
\gamma A^*{}^2+\omega A^*\!+\eta^*I.
\end{eqnarray*}
The sequence is uniquely determined by the pair $A,A^*$ provided
the dimension of $V$ is at least $4$.
\end{theorem}

\noindent 
The following theorem is a kind of converse to Theorem
 \ref{lptheorem}.

\begin{theorem}
\cite{aw}
Let $V$ denote a vector space over $\K$ with finite positive
dimension. Let $A:V\to V$ and $A^*:V\to V$ denote linear
transformations. Suppose that:
\begin{itemize}
\item There exists a sequence of scalars
$\beta,\gamma,\gamma^*,\varrho,\varrho^*,\omega,\eta,\eta^*$ taken
from $\K$ which satisfies {\rm (\ref{askwil1})}, {\rm
(\ref{askwil2})}.
\item $q$ is not a root of unity, where $q+q^{-1}=\beta$.
\item Each of $A$ and $A^*$ is multiplicity-free.
\item There does not exist a subspace $W\subseteq V$ such that
$W\not=0$,
$W\not=V$,
$AW\subseteq W$,
$A^*W\subseteq W$.
\end{itemize}
Then $A,A^*$ is a Leonard pair on $V$.
\end{theorem}


\section{Leonard pairs and the Lie algebra $sl_2$}

\medskip
\noindent
In this section we assume
the field $\K$ is algebraically closed 
with characteristic zero.

\medskip
\noindent We recall the Lie algebra $sl_2=sl_2(\K)$. 
This algebra has a basis  $e,f,h$ satisfying
\beast
&&\lbrack h,e \rbrack  = 2e, \qquad  
\lbrack h,f \rbrack  = -2f, 
\qquad
\lbrack e,f \rbrack  = h,
\eeast
where $\lbrack\,,\, \rbrack $ denotes the Lie bracket.

\medskip
\noindent 
We recall the irreducible finite dimensional modules for $sl_2$.
\begin{lemma} \cite[p. 102]{Kassel} 
\label{lemvd}
There exists a family 
\begin{equation}
\label{sl2modlist}
V_d  \qquad \qquad d = 0,1,2\ldots
\end{equation}
of irreducible finite dimensional $sl_2$-modules with the following
properties. The module $V_d$ has a basis $v_0, v_1, \ldots, v_d$
satisfying $h v_i =(d-2i)v_i$ for $0 \leq i \leq d$,
$fv_i = (i+1)v_{i+1}$ for $0 \leq i \leq d-1$, $fv_d=0$, 
$ev_i = (d-i+1)v_{i-1}$ for $1 \leq i \leq d$,  $ev_0=0$.
Every irreducible finite dimensional $sl_2$-module
is isomorphic to exactly one of the modules 
in line (\ref{sl2modlist}).
\end{lemma}


\begin{theorem} 
\label{ex:slgen}
\cite[Ex. 1.5]{TD00}
Let $A$ and $A^*$ denote semi-simple 
elements in $sl_2$ and assume $sl_2$ is generated by
these elements. 
Let $V$ denote an irreducible finite dimensional
 module for $sl_2$. Then
the pair $A, A^*$ acts on $V$ as a Leonard  pair.
\end{theorem}

\noindent We remark the Leonard pairs in Theorem 
\ref{ex:slgen} correspond to the  Krawtchouk
polynomials
 \cite{KoeSwa}.

\section{Leonard pairs and $U_q(sl_2)$}


\medskip
\noindent In this section we assume $\K$ is algebraically closed.
We fix a nonzero scalar $q\in \K$ which is not a root of unity. 
We recall the quantum algebra
 $U_q(sl_2)$.

\begin{definition} 
\cite[p.122]{Kassel}
\label{def:uqsl2}
Let $U_{q}(sl_2)$ denote the associative $\K$-algebra 
with 1 generated by symbols $ e,  f, k, k^{-1}$ subject
to the relations
\beast
kk^{-1} = k^{-1}k= 1,
\eeast
\beast
 ke = q^2 ek,\qquad\qquad  kf = q^{-2}fk,
\eeast
\beast
ef - fe  = {{k-k^{-1}}\over {q - q^{-1}}}.
\eeast

\end{definition}
\noindent We recall the irreducible finite dimensional modules
for 
 $U_{q}(sl_2)$. We use the following notation.
\beast
\lbrack n \rbrack_q = 
{{q^n - q^{-n}}\over {q-q^{-1}}} \qquad 
\qquad n \in \Z .
\eeast


\begin{lemma} \cite[p. 128]{Kassel}
\label{lem:uqmods}
With reference to Definition \ref{def:uqsl2},
there exists a family 
\begin{eqnarray}
V_{\varepsilon,d} \qquad \quad 
\varepsilon \in \lbrace 1,-1\rbrace, \qquad \quad  d = 0,1,2\ldots
\label{eq:uqmods}
\end{eqnarray}
of 
 irreducible finite dimensional 
 $U_{q}(sl_2)$-modules
 with the following properties.
The module 
$V_{\varepsilon,d}$ has a basis $u_0, u_1, \ldots, u_d$
satisfying $ku_i=\varepsilon q^{d-2i}u_i$ for
$0 \leq i \leq d$, $fu_i=\lbrack i+1\rbrack_q u_{i+1}$
for $0 \leq i \leq d-1$, $fu_d=0$, 
 $eu_i=\varepsilon\lbrack d-i+1\rbrack_q u_{i-1}$
for $1 \leq i \leq d$, $eu_0=0$. 
Every irreducible finite dimensional
 $U_{q}(sl_2)$-module is isomorphic to exactly one of  
the modules $V_{\varepsilon,d}$.
(Referring to line
(\ref{eq:uqmods}), if 
$\K$ has characteristic 2
we interpret the set $\lbrace 1,-1 \rbrace $ as having 
a single element.) 
\end{lemma}

%\medskip
%\noindent
%We use the above $U_q(sl_2)$-modules to get Leonard pairs.
%The following result was proved by the present author in
%\cite{Terint}
%and is implicit in the results of Koelink and Van der Jeugt
%\cite{cite38}, \cite{cite39}. For related results see
%\cite{cite37},
%\cite{cite40},
%\cite{Rosengren},
%\cite{qSerre}. 

\begin{theorem}
\cite{cite38}, \cite{cite39},
\cite{Terint}
\label{ex:uqex}
Referring to Definition
\ref{def:uqsl2} and Lemma 
\ref{lem:uqmods},
let  $\alpha, \beta $ denote nonzero scalars in  $\K $ and 
 define $A$, $A^*$ as follows.
\begin{eqnarray*}
A= \alpha f + {{ k}\over {q-q^{-1}}},
\qquad \qquad 
A^*=\beta e + {{k^{-1}}\over {q-q^{-1}}}.
\label{eq:abdef}
\end{eqnarray*}
Let $d$ denote a nonnegative integer and choose 
$\varepsilon \in \lbrace 1, -1 \rbrace $.
 Then the pair $A, A^*$ acts on $V_{\varepsilon,d}$ as a Leonard pair
provided $\varepsilon \alpha \beta $ is not among
$q^{d-1}, q^{d-3},\ldots, q^{1-d}$. 
\end{theorem}

\noindent 
We remark  
the Leonard pairs in 
Theorem
\ref{ex:uqex}
correspond to the quantum $q$-Krawtchouk
polynomials \cite{KoeSwa}, 
\cite{cite37}. 

\section{Leonard pairs in  combinatorics} 


Leonard pairs arise in many branches of combinatorics.
For instance they arise in
the theory of partially ordered sets (posets).
We illustrate this with a poset called the  
subspace lattice $L_n(q)$.

\medskip
\noindent 
In this section we assume our field $\K$ is the field $\C$ of
complex numbers.

\medskip
\noindent To define the subspace lattice
 we introduce a second field. 
Let $GF(q)$ denote a finite field of order $q$.
Let $n$ denote a positive integer
and let $W$ denote an $n$-dimensional vector space over $GF(q)$.
Let $P$ denote the set consisting of all subspaces of $W$.
The set $P$, together with the containment relation, is a poset
called $L_n(q)$. 

\medskip
\noindent
Using $L_n(q)$ we obtain a family of Leonard pairs as follows.
Let  $\C P$  denote
the vector space over $\C$ consisting of all formal $\C$-linear combinations
of elements of $P$. 
We observe $P$ is a basis for  
 $\C P$ so
the dimension of $\C P$ is equal to the cardinality of P.

\medskip
\noindent
We define  three linear transformations on $\C P$. We call these
$K$, $R$ (for ``raising''), $L$ (for ``lowering'').

\medskip
\noindent
We begin with $K$. For all $x \in P$,
\beast  
	  K x =   q^{{n/2} - dim\, x} x.
	 % \qquad \qquad    (x \in P).
\eeast
Apparently each element of $P$ is an eigenvector for $K$.

\medskip
\noindent
To define $R$ and $L$ we use the following notation. 
For $x, y \in P$ we say $y$ {\it covers}  $x$
whenever (i) $x \subseteq y$ and  (ii) $\hbox{dim}\, y= 1+\hbox{dim}\,x$.

\medskip
\noindent The maps 
$R$ and $L$ are defined as follows. For all $x \in  P$, 
\beast
       %    R x = \sum_{{y \in P}\atop{y \;covers\; x}} y.
           R x = \sum_{y \;covers \; x} y.
\eeast  
%where the sum is over all elements y of P that cover x.
Similarly
%for all $x \in P$, 
 \beast 
         %  L x =q^{(1-n)/2} \sum_{{y \in P}\atop{x \;\hbox{covers}\; y}} y.
           L x =q^{(1-n)/2} \sum_{x \; covers \; y} y.
\eeast
%where the sum is over all elements y of P that are covered by x.
(The scalar $q^{(1-n)/2}$ is included for aesthetic reasons.)

\medskip
\noindent We consider the properties of $K, R, L$.
From the construction we find $K^{-1}$ exists.
By combinatorial counting we  verify 
\beast
     &&     K L = q L K,   \qquad \qquad \qquad   K R = q^{-1} R K,
     \\
       && \qquad          L R - R L = \frac{K - K^{-1}}{q^{1/2} - q^{-1/2}}.
\eeast
We recognize these equations. They are the defining relations for 
$U_{q^{1/2}}(sl_2)$.  Apparently $K$, $R$, $L$ turn  $\C P$ into
a module for $U_{q^{1/2}}(sl_2)$. 

\medskip
\noindent
We now see how to get Leonard pairs from $L_n(q)$.
Let  $\alpha, \beta$ denote nonzero complex scalars and 
define $A$, $A^*$ as follows.
\beast
                  A =  \alpha R    + \frac{K}{q^{1/2}-q^{-1/2}},
        \qquad \qquad  A^* =  \beta  L    + \frac{K^{-1}}{q^{1/2}-q^{-1/2}}.
\eeast
To avoid 
degenerate situations we assume  
$\alpha \beta $  is not among $q^{(n-1)/2}, q^{(n-3)/2},\ldots, q^{(1-n)/2}$.

\medskip
\noindent The  $U_{q^{1/2}}(sl_2)$-module $\C P$ is 
completely reducible \cite[p. 144]{Kassel}.
In other words $\C P$ 
is a direct sum of irreducible
$U_{q^{1/2}}(sl_2)$-modules. On
each irreducible module in this sum
the pair $A, A^*$ acts as a Leonard pair. This follows from
Theorem 
\ref{ex:uqex}.

\medskip
\noindent
We just saw how the subspace lattice gives Leonard pairs. 
It is implicit in \cite{uniform} that
the following posets give 
Leonard pairs in a similar fashion:
the subset lattice,
the Hamming semi-lattice,
the attenuated spaces,
and the classical polar spaces. 
Definitions of these posets can be found in 
 \cite{uniform}.

\section{Further reading}

We mention some additional topics which are related 
to Leonard pairs.

\medskip
\noindent 
Earlier in this paper we obtained 
Leonard pairs from the irreducible finite dimensional modules
for the Lie algebra $sl_2$ and the quantum algebra $U_q(sl_2)$.
We cite some other 
algebras whose modules are related to Leonard pairs.
These are the Askey-Wilson algebra
\cite{GYZnature},
 \cite{GYLZmut},
\cite{GYZTwisted},
\cite{GYZlinear},
\cite{GYZspherical},
\cite{Zhidd}, 
\cite{ZheCart},
\cite{Zhidden}, the Onsager algebra
\cite{CKOnsn},
\cite{DateRoan2},
\cite{Dav},
\cite{Dolgra},
and the Tridiagonal algebra
\cite{TD00},
\cite{LS99},
\cite{qSerre}.

\medskip
\noindent 
We discussed how
certain classical posets give Leonard pairs.
Another combinatorial object which gives Leonard pairs is
 a  $P$- and $Q$-polynomial
association scheme \cite{BanIto}, \cite{bcn}, 
 \cite{TersubI}.
Leonard pairs have been used to describe
certain irreducible modules for the subconstituent algebra of
 these schemes
\cite{Cau}, \cite{CurNom}, \cite{Curspin}, 
\cite{TD00},
 \cite{TersubI}.

\medskip
\noindent 
The topic of Leonard pairs is closely related to the  work of 
Gr\"unbaum and Haine on the 
 ``bispectral problem''
\cite{GH7},
\cite{GH6}.
See
\cite{GH4},
\cite{GH5},
\cite{GH1}, 
\cite{GH3},
\cite{GH2} 
for related work. 

{\small
\begin{thebibliography}{10}


\bibitem{CKOnsn}
C.~Ahn and K.~Shigemoto,
\newblock Onsager algebra and integrable lattice models,
\newblock {\em Modern Phys. Lett. A} {\bf 6(38)} (1991) 3509--3515.

\bibitem{Ask}
G.~Andrews, R.~Askey, and R.~Roy,
\newblock  {\em Special functions},
\newblock Cambridge University Press, Cambridge, 1999. 

\bibitem{BanIto}
E.~Bannai and T.~Ito,
\newblock {\em Algebraic Combinatorics {I}: Association Schemes},
\newblock Benjamin/Cummings, London, 1984.

\bibitem{bcn}
A.~E. Brouwer, A.~M. Cohen, and A.~Neumaier,
\newblock {\em Distance-Regular Graphs},
\newblock Springer-Verlag, Berlin, 1989.

\bibitem{Cau}
J.~S. Caughman~{I}{V},
\newblock The {T}erwilliger algebras of bipartite ${P}$- and ${Q}$-polynomial
  schemes,
\newblock {\em Discrete Math.} {\bf 196} (1999) 65--95.


\bibitem{CurNom}
B.~Curtin and K.~Nomura,
\newblock Distance-regular graphs related to the quantum enveloping algebra of
  $sl(2)$,
\newblock {\em J. Algebraic Combin.} 
{\bf 12} (2000) 25--36.

\bibitem{Curspin}
B.~Curtin,
\newblock Distance-regular graphs which support a spin model are thin, in:
Proc.  16th British Combinatorial Conference
(London, 1997),
\newblock {\em Discrete Math.} {\bf 197/198} (1999) 205--216.

\bibitem{DateRoan2}
E.~Date and S.S. Roan,
\newblock The structure of quotients of the {O}nsager algebra by closed ideals,
\newblock {\em J. Phys. A: Math. Gen.} {\bf 33} (2000) 3275--3296.

\bibitem{Dav}
B.~Davies,
\newblock Onsager's algebra and the {D}olan-{G}rady condition in the
  non-self-dual case,
\newblock {\em J. Math. Phys.} {\bf 32} (1991) 2945--2950.

\bibitem{Dolgra}
L.~Dolan and M.~Grady,
\newblock Conserved charges from self-duality,
\newblock {\em Phys. Rev. D (3)} {\bf 25} (1982) 1587--1604.

\bibitem{GR}
G.~Gasper and M.~Rahman,
\newblock {\em Basic hypergeometric series}.
\newblock Encyclopedia of Mathematics and its Applications, 35,
\newblock Cambridge University Press, Cambridge, 1990.


\bibitem{GYZnature}
Ya.~A. Granovski{\u\i} and A.~S. Zhedanov,
\newblock Nature of the symmetry group of the $6j$-symbol,
\newblock {\em Zh. \`Eksper. Teoret. Fiz.}  {\bf 94} (1988) 49--54.

\bibitem{GYLZmut}
Ya.~I. Granovski{\u\i}, I.~M. Lutzenko, and A.~S. Zhedanov,
\newblock Mutual integrability, quadratic algebras, and dynamical symmetry,
\newblock {\em Ann. Physics} {\bf 217} (1992) 1--20.

\bibitem{GYZTwisted}
Ya.~I. Granovski{\u\i} and A.~S. Zhedanov,
\newblock ``{T}wisted'' {C}lebsch-{G}ordan coefficients for ${\rm {s}{u}}\sb
  q(2)$,
\newblock {\em J. Phys. A} {\bf 25} (1992) L1029--L1032.

\bibitem{GYZlinear}
Ya.~I. Granovski{\u\i} and A.~S. Zhedanov,
\newblock Linear covariance algebra for ${\rm {s}{l}}\sb q(2)$,
\newblock {\em J. Phys. A} {\bf 26} (1993) L357--L359.

\bibitem{GYZspherical}
Ya.~I. Granovski{\u\i} and A.~S. Zhedanov,
\newblock Spherical $q$-functions,
\newblock {\em J. Phys. A} {\bf 26} (1993) 4331--4338.

\bibitem{GH4}
F.~A.~ Gr{\"u}nbaum,
\newblock Some bispectral musings, in: 
\newblock  {\em The bispectral problem (Montreal, PQ, 1997)},
  Amer. Math. Soc., Providence, RI, 1998, 
 pp. 31--45.

\bibitem{GH5}
F.~A.~Gr{\"u}nbaum and L.~Haine,
\newblock Bispectral {D}arboux transformations: an extension of the {K}rall
  polynomials,
\newblock {\em Internat. Math. Res. Notices} {\bf 8} (1997) 359--392.

\bibitem{GH7}
F.~A.~Gr{\"u}nbaum and L.~Haine,
\newblock The $q$-version of a theorem of {B}ochner,
\newblock {\em J. Comput. Appl. Math.}  {\bf 68} (1996) 103--114.

\bibitem{GH6}
F.~A.~Gr{\"u}nbaum and L.~Haine,
\newblock Some functions that generalize the {A}skey-{W}ilson polynomials,
\newblock {\em Comm. Math. Phys.} {\bf 184} (1997) 173--202.

\bibitem{GH1}
F.~A.~Gr{\"u}nbaum and L.~Haine,
\newblock On a $q$-analogue of the string equation and a generalization of the
  classical orthogonal polynomials, in: 
\newblock  {\em Algebraic methods and $q$-special functions (Montr\'eal, QC,
  1996)}
   Amer. Math. Soc., Providence, RI, 1999, pp. 
  171--181.

\bibitem{GH3}
F.~A.~Gr{\"u}nbaum and L.~Haine,
\newblock The {W}ilson bispectral involution: some elementary examples, in:
\newblock  {\em Symmetries and integrability of difference equations
  (Canterbury, 1996)},
   Cambridge Univ. Press, Cambridge, 1999, 
  pp. 353--369. 

\bibitem{GH2}
F.~A.~Gr{\"u}nbaum, L.~Haine, and E.~Horozov,
\newblock Some functions that generalize the {K}rall-{L}aguerre polynomials,
\newblock {\em J. Comput. Appl. Math.} {\bf 106} (1999) 271--297.

\bibitem{TD00}
T.~Ito, K.~Tanabe, and P.~Terwilliger,
\newblock Some algebra related to ${P}$- and ${Q}$-polynomial association
  schemes,  in:
\newblock {\em Codes and Association Schemes (Piscataway NJ, 1999)}, Amer.
Math. Soc., Providence RI, 2001, pp.
     167--192.

\bibitem{shape}
T.~Ito and P.~Terwilliger.
\newblock The shape of a tridiagonal pair.
\newblock {\em J. Pure Appl. Algebra}, submitted.
Available at
\newblock{ 
\tt  http://front.math.ucdavis.edu/math.QA/0304244
}



\bibitem{Kassel}
C.~Kassel,
\newblock {\em Quantum groups},
\newblock Springer-Verlag,
New York,
    1995.

\bibitem{KoeSwa}
R.~Koekoek and R.~F.~Swarttouw,
\newblock {\em The Askey scheme of hypergeometric orthogonal
polyomials and its
  $q$-analog}, report 98-17, Delft University of Technology, The
  Netherlands, 1998.
  Available at
 \newblock{ \tt http://aw.twi.tudelft.nl/{\~{}}koekoek/research.html} 

\bibitem{Koelink3}
H.~T. Koelink,
\newblock Askey-{W}ilson polynomials and the quantum ${\rm {s}{u}}(2)$ group:
  survey and applications,
\newblock {\em Acta Appl. Math.} {\bf 44} (1996) 295--352.

\bibitem{cite37} 
H.~T. Koelink,
\newblock $q$-Krawtchouk polynomials as spherical functions on the Hecke
algebra of type $B$,
\newblock {\em Trans. Amer. Math. Soc.} {\bf 352} (2000) 4789--4813.

\bibitem{cite38}
H.~T. Koelink and J.~Van der Jeugt,
\newblock  Convolutions for orthogonal polynomials from Lie and 
quantum algebra representations,
\newblock {\em SIAM J. Math. Anal.} {\bf 29} (1998) 794--822.


\bibitem{cite39}
H.~T. Koelink and J.~Van der Jeugt,
\newblock  Bilinear generating functions for orthogonal polynomials,
\newblock {\em Constr. Approx.} {\bf 15} (1999) 481--497.

\bibitem{cite40}
T.~H.~Koornwinder,
\newblock Askey-Wilson polynomials as zonal spherical functions on
the su(2) quantum group,
\newblock {\em SIAM J. Math. Anal.} {\bf 24} (1993) 795--813.

\bibitem{Leon}
D.~Leonard,
\newblock Orthogonal polynomials, duality, and association
schemes,
\newblock {\em SIAM J. Math. Anal.} {\bf 13} (1982) 656--663.

\bibitem{Rosengren}
H.~Rosengren,
\newblock {\em Multivariable orthogonal polynomials as coupling
coefficients for Lie and quantum algebra representations.}
\newblock Centre for Mathematical Sciences, Lund University, Sweden,
1999.


\bibitem{uniform}
P.~Terwilliger,
\newblock The incidence algebra of a uniform  poset,
\newblock {\em Math and its applications} {\bf 20} (1990) 193--212.


\bibitem{TersubI}
P.~Terwilliger,
\newblock The subconstituent algebra of an association scheme, 
\newblock {\em J. Algebraic Combin.} {\bf 1} (1992) 363--388.
    

\bibitem{LS99}
P.~Terwilliger.
\newblock Two linear transformations each tridiagonal with respect to an
  eigenbasis of the other.
  \newblock {\em Linear Algebra Appl.}  {\bf 330} (2001), 149--203.

  \bibitem{qSerre}
  P.~Terwilliger.
  \newblock Two relations that generalize the $q$-Serre relations and the
  Dolan-Grady relations. In
  \newblock {\em  Physics and
  Combinatorics 1999 (Nagoya)}, 377--398, World Scientific Publishing,
   River Edge, NJ, 2001

   \bibitem{LS24}
   P.~Terwilliger.
   \newblock  Leonard pairs from 24 points of view.
   \newblock {\em Rocky Mountain J. Math.} {\bf 32}(2) (2002), 827--888.

   \bibitem{conform}
   P.~Terwilliger.
   \newblock Two linear transformations each tridiagonal with respect to an
     eigenbasis of the other; the $TD$-$D$ and the $LB$-$UB$ canonical form.
\newblock {\em J. Algebra}. Submitted.
Available at
\newblock{ 
\tt  http://front.math.ucdavis.edu/math.RA/0304077
}


    \bibitem{lsint}
    P.~Terwilliger.
    \newblock Introduction to Leonard pairs.
    \newblock {OPSFA Rome 2001}.
    \newblock{\em J. Comput. Appl. Math.} {\bf 153}(2) (2003),
    463--475.

\bibitem{Terint}
P.~Terwilliger.
\newblock Introduction to {L}eonard pairs and
  {L}eonard systems.
  \newblock {\em S\=urikaisekikenky\=usho K\=oky\=uroku},
 \newblock
 (1109):67--79, 1999.
\newblock Algebraic combinatorics  (Kyoto, 1999).

\bibitem{TLT:split}
P.~Terwilliger.
\newblock Two linear transformations each tridiagonal with respect to an
  eigenbasis of the other; comments on the split decomposition.
\newblock {\em Indag. Math.} Submitted.
Available at
\newblock{ 
\tt  http://front.math.ucdavis.edu/math.RA/0306290
}



 \bibitem{TLT:array}
 P.~Terwilliger.
 \newblock Two linear transformations each tridiagonal with respect to an
   eigenbasis of the other; comments on the parameter array.
 \newblock {\em Geometric and Algebraic Combinatorics 2,
 Oisterwijk, The Netherlands 2002}.
  \newblock Submitted.
Available at
\newblock{
\tt  http://front.math.ucdavis.edu/math.RA/0306291
}

\bibitem{qrac}
P.~Terwilliger.
\newblock Leonard pairs and the $q$-Racah polynomials.
\newblock {\em Linear Algebra Appl.} Submitted.

\bibitem{aw}
P.~Terwilliger and R.~Vidunas.
\newblock Leonard pairs and the Askey-Wilson relations.
\newblock {\em J. Algebra Appl.} Submitted.
Available at
\newblock{ 
\tt  http://front.math.ucdavis.edu/math.QA/0305356
}



\bibitem{Zhidd}
A.~S. Zhedanov,
\newblock ``{H}idden symmetry'' of {A}skey-{W}ilson polynomials,
\newblock {\em Teoret. Mat. Fiz.} {\bf 89} (1991) 190--204.

\bibitem{ZheCart}
A.~S. Zhedanov,
\newblock Quantum ${\rm {s}{u}}\sb q(2)$ algebra: ``{C}artesian'' version and
  overlaps,
\newblock {\em Modern Phys. Lett. A} {\bf 7} (1992) 1589--1593.

\bibitem{Zhidden}
A.~S. Zhedanov,
\newblock Hidden symmetry algebra and overlap coefficients for two ring-shaped
  potentials,
\newblock {\em J. Phys. A} {\bf 26} (1993) 4633--4641.

\end{thebibliography}
}

\noindent Paul Terwilliger, Department of Mathematics, University of
Wisconsin, 480 Lincoln Drive, Madison, Wisconsin, 53706, USA \hfil\break
Email: terwilli@math.wisc.edu \hfil\break

%}}
\end{document}




math0307063.extracted_bib
\begin{thebibliography}{10}


\bibitem{CKOnsn}
C.~Ahn and K.~Shigemoto,
\newblock Onsager algebra and integrable lattice models,
\newblock {\em Modern Phys. Lett. A} {\bf 6(38)} (1991) 3509--3515.

\bibitem{Ask}
G.~Andrews, R.~Askey, and R.~Roy,
\newblock  {\em Special functions},
\newblock Cambridge University Press, Cambridge, 1999. 

\bibitem{BanIto}
E.~Bannai and T.~Ito,
\newblock {\em Algebraic Combinatorics {I}: Association Schemes},
\newblock Benjamin/Cummings, London, 1984.

\bibitem{bcn}
A.~E. Brouwer, A.~M. Cohen, and A.~Neumaier,
\newblock {\em Distance-Regular Graphs},
\newblock Springer-Verlag, Berlin, 1989.

\bibitem{Cau}
J.~S. Caughman~{I}{V},
\newblock The {T}erwilliger algebras of bipartite ${P}$- and ${Q}$-polynomial
  schemes,
\newblock {\em Discrete Math.} {\bf 196} (1999) 65--95.


\bibitem{CurNom}
B.~Curtin and K.~Nomura,
\newblock Distance-regular graphs related to the quantum enveloping algebra of
  $sl(2)$,
\newblock {\em J. Algebraic Combin.} 
{\bf 12} (2000) 25--36.

\bibitem{Curspin}
B.~Curtin,
\newblock Distance-regular graphs which support a spin model are thin, in:
Proc.  16th British Combinatorial Conference
(London, 1997),
\newblock {\em Discrete Math.} {\bf 197/198} (1999) 205--216.

\bibitem{DateRoan2}
E.~Date and S.S. Roan,
\newblock The structure of quotients of the {O}nsager algebra by closed ideals,
\newblock {\em J. Phys. A: Math. Gen.} {\bf 33} (2000) 3275--3296.

\bibitem{Dav}
B.~Davies,
\newblock Onsager's algebra and the {D}olan-{G}rady condition in the
  non-self-dual case,
\newblock {\em J. Math. Phys.} {\bf 32} (1991) 2945--2950.

\bibitem{Dolgra}
L.~Dolan and M.~Grady,
\newblock Conserved charges from self-duality,
\newblock {\em Phys. Rev. D (3)} {\bf 25} (1982) 1587--1604.

\bibitem{GR}
G.~Gasper and M.~Rahman,
\newblock {\em Basic hypergeometric series}.
\newblock Encyclopedia of Mathematics and its Applications, 35,
\newblock Cambridge University Press, Cambridge, 1990.


\bibitem{GYZnature}
Ya.~A. Granovski{\u\i} and A.~S. Zhedanov,
\newblock Nature of the symmetry group of the $6j$-symbol,
\newblock {\em Zh. \`Eksper. Teoret. Fiz.}  {\bf 94} (1988) 49--54.

\bibitem{GYLZmut}
Ya.~I. Granovski{\u\i}, I.~M. Lutzenko, and A.~S. Zhedanov,
\newblock Mutual integrability, quadratic algebras, and dynamical symmetry,
\newblock {\em Ann. Physics} {\bf 217} (1992) 1--20.

\bibitem{GYZTwisted}
Ya.~I. Granovski{\u\i} and A.~S. Zhedanov,
\newblock ``{T}wisted'' {C}lebsch-{G}ordan coefficients for ${\rm {s}{u}}\sb
  q(2)$,
\newblock {\em J. Phys. A} {\bf 25} (1992) L1029--L1032.

\bibitem{GYZlinear}
Ya.~I. Granovski{\u\i} and A.~S. Zhedanov,
\newblock Linear covariance algebra for ${\rm {s}{l}}\sb q(2)$,
\newblock {\em J. Phys. A} {\bf 26} (1993) L357--L359.

\bibitem{GYZspherical}
Ya.~I. Granovski{\u\i} and A.~S. Zhedanov,
\newblock Spherical $q$-functions,
\newblock {\em J. Phys. A} {\bf 26} (1993) 4331--4338.

\bibitem{GH4}
F.~A.~ Gr{\"u}nbaum,
\newblock Some bispectral musings, in: 
\newblock  {\em The bispectral problem (Montreal, PQ, 1997)},
  Amer. Math. Soc., Providence, RI, 1998, 
 pp. 31--45.

\bibitem{GH5}
F.~A.~Gr{\"u}nbaum and L.~Haine,
\newblock Bispectral {D}arboux transformations: an extension of the {K}rall
  polynomials,
\newblock {\em Internat. Math. Res. Notices} {\bf 8} (1997) 359--392.

\bibitem{GH7}
F.~A.~Gr{\"u}nbaum and L.~Haine,
\newblock The $q$-version of a theorem of {B}ochner,
\newblock {\em J. Comput. Appl. Math.}  {\bf 68} (1996) 103--114.

\bibitem{GH6}
F.~A.~Gr{\"u}nbaum and L.~Haine,
\newblock Some functions that generalize the {A}skey-{W}ilson polynomials,
\newblock {\em Comm. Math. Phys.} {\bf 184} (1997) 173--202.

\bibitem{GH1}
F.~A.~Gr{\"u}nbaum and L.~Haine,
\newblock On a $q$-analogue of the string equation and a generalization of the
  classical orthogonal polynomials, in: 
\newblock  {\em Algebraic methods and $q$-special functions (Montr\'eal, QC,
  1996)}
   Amer. Math. Soc., Providence, RI, 1999, pp. 
  171--181.

\bibitem{GH3}
F.~A.~Gr{\"u}nbaum and L.~Haine,
\newblock The {W}ilson bispectral involution: some elementary examples, in:
\newblock  {\em Symmetries and integrability of difference equations
  (Canterbury, 1996)},
   Cambridge Univ. Press, Cambridge, 1999, 
  pp. 353--369. 

\bibitem{GH2}
F.~A.~Gr{\"u}nbaum, L.~Haine, and E.~Horozov,
\newblock Some functions that generalize the {K}rall-{L}aguerre polynomials,
\newblock {\em J. Comput. Appl. Math.} {\bf 106} (1999) 271--297.

\bibitem{TD00}
T.~Ito, K.~Tanabe, and P.~Terwilliger,
\newblock Some algebra related to ${P}$- and ${Q}$-polynomial association
  schemes,  in:
\newblock {\em Codes and Association Schemes (Piscataway NJ, 1999)}, Amer.
Math. Soc., Providence RI, 2001, pp.
     167--192.

\bibitem{shape}
T.~Ito and P.~Terwilliger.
\newblock The shape of a tridiagonal pair.
\newblock {\em J. Pure Appl. Algebra}, submitted.
Available at
\newblock{ 
\tt  http://front.math.ucdavis.edu/math.QA/0304244
}



\bibitem{Kassel}
C.~Kassel,
\newblock {\em Quantum groups},
\newblock Springer-Verlag,
New York,
    1995.

\bibitem{KoeSwa}
R.~Koekoek and R.~F.~Swarttouw,
\newblock {\em The Askey scheme of hypergeometric orthogonal
polyomials and its
  $q$-analog}, report 98-17, Delft University of Technology, The
  Netherlands, 1998.
  Available at
 \newblock{ \tt http://aw.twi.tudelft.nl/{\~{}}koekoek/research.html} 

\bibitem{Koelink3}
H.~T. Koelink,
\newblock Askey-{W}ilson polynomials and the quantum ${\rm {s}{u}}(2)$ group:
  survey and applications,
\newblock {\em Acta Appl. Math.} {\bf 44} (1996) 295--352.

\bibitem{cite37} 
H.~T. Koelink,
\newblock $q$-Krawtchouk polynomials as spherical functions on the Hecke
algebra of type $B$,
\newblock {\em Trans. Amer. Math. Soc.} {\bf 352} (2000) 4789--4813.

\bibitem{cite38}
H.~T. Koelink and J.~Van der Jeugt,
\newblock  Convolutions for orthogonal polynomials from Lie and 
quantum algebra representations,
\newblock {\em SIAM J. Math. Anal.} {\bf 29} (1998) 794--822.


\bibitem{cite39}
H.~T. Koelink and J.~Van der Jeugt,
\newblock  Bilinear generating functions for orthogonal polynomials,
\newblock {\em Constr. Approx.} {\bf 15} (1999) 481--497.

\bibitem{cite40}
T.~H.~Koornwinder,
\newblock Askey-Wilson polynomials as zonal spherical functions on
the su(2) quantum group,
\newblock {\em SIAM J. Math. Anal.} {\bf 24} (1993) 795--813.

\bibitem{Leon}
D.~Leonard,
\newblock Orthogonal polynomials, duality, and association
schemes,
\newblock {\em SIAM J. Math. Anal.} {\bf 13} (1982) 656--663.

\bibitem{Rosengren}
H.~Rosengren,
\newblock {\em Multivariable orthogonal polynomials as coupling
coefficients for Lie and quantum algebra representations.}
\newblock Centre for Mathematical Sciences, Lund University, Sweden,
1999.


\bibitem{uniform}
P.~Terwilliger,
\newblock The incidence algebra of a uniform  poset,
\newblock {\em Math and its applications} {\bf 20} (1990) 193--212.


\bibitem{TersubI}
P.~Terwilliger,
\newblock The subconstituent algebra of an association scheme, 
\newblock {\em J. Algebraic Combin.} {\bf 1} (1992) 363--388.
    

\bibitem{LS99}
P.~Terwilliger.
\newblock Two linear transformations each tridiagonal with respect to an
  eigenbasis of the other.
  \newblock {\em Linear Algebra Appl.}  {\bf 330} (2001), 149--203.

  \bibitem{qSerre}
  P.~Terwilliger.
  \newblock Two relations that generalize the $q$-Serre relations and the
  Dolan-Grady relations. In
  \newblock {\em  Physics and
  Combinatorics 1999 (Nagoya)}, 377--398, World Scientific Publishing,
   River Edge, NJ, 2001

   \bibitem{LS24}
   P.~Terwilliger.
   \newblock  Leonard pairs from 24 points of view.
   \newblock {\em Rocky Mountain J. Math.} {\bf 32}(2) (2002), 827--888.

   \bibitem{conform}
   P.~Terwilliger.
   \newblock Two linear transformations each tridiagonal with respect to an
     eigenbasis of the other; the $TD$-$D$ and the $LB$-$UB$ canonical form.
\newblock {\em J. Algebra}. Submitted.
Available at
\newblock{ 
\tt  http://front.math.ucdavis.edu/math.RA/0304077
}


    \bibitem{lsint}
    P.~Terwilliger.
    \newblock Introduction to Leonard pairs.
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\bibitem{Terint}
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\bibitem{TLT:split}
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\newblock Two linear transformations each tridiagonal with respect to an
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\newblock{ 
\tt  http://front.math.ucdavis.edu/math.RA/0306290
}



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\newblock{
\tt  http://front.math.ucdavis.edu/math.RA/0306291
}

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\newblock{ 
\tt  http://front.math.ucdavis.edu/math.QA/0305356
}



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\end{thebibliography}