Paul Terwilliger, University of Wisconsin, Madison, U.S.A.
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.

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