e-mail: terwilli@math.wisc.edu

**Tridiagonal Pairs**

**Abstract**

We consider the following situation in linear algebra: Let $F$ denote
an arbitrary field, and let $V$ denote a vector space over $F$ with
finite positive dimension. Let $A$ and $B$ denote linear transformations
from $V$ to $V$ that satisfy the following (i)-(iv):

(i) $A$ and $B$ are both diagonalizable on $V$

(ii) There exists an ordering $V_0, V_1,\ldots V_d$ of the maximal
eigenspaces of $A$ such that $BV_i$ is contained in

$V_{i-1} + V_i + V_{i+1}$ for $0 \leq i \leq d$.

(iii) There exists an ordering $V^*_0, V^*_1, \ldots V^*_e$ of
the maximal eigenspaces of $B$ such that $AV^*_i$ is contained in
$V^*_{i-1}+V^*_i+V^*_{i+1}$ for $0 \leq i \leq e$.

(iv) There is no subspace $W$ of $V$ that is invariant under
both $A$ and $B$, other than $W=0$ and $W=V$.

We call such a pair A,B a {\it tridiagonal pair}. This is a generalization
of the notion of a Leonard pair. By a {\it Leonard pair}, we mean a tridiagonal
pair for which the eigenspaces $V_i$ and $V^*_i$ all have dimension 1.
In an earlier paper, we obtained a natural 1--1 correspondance between
the Leonard pairs and the finite length polynomial sequences of the Askey-Scheme.
The most general polynomials of this sort are the $q$-Racah polynomials.
In this paper, we begin a classification of the general tridiagonal pairs.
We introduce a certain infinite dimensional algebra which we call the {\it
tridiagonal algebra}. This is a generalization of the Askey-Wilson algebra
of Zhedanov. We show every tridiagonal pair is associated with an irreducible
finite dimensional representation of a Tridiagonal algebra.