Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations

Abstract

We define an algebra on two generators which we call the Tridiagonal algebra, and we consider its irreducible modules. The algebra is defined as follows. Let K denote a field, and let $\beta, \gamma, \gamma^*, \varrho, \varrho^*$ denote a sequence of scalars taken from K. The corresponding Tridiagonal algebra $T$ is the associative K-algebra with 1 generated by two symbols $A$, $A^*$ subject to the relations (i) \lbrack A,A^2A^*-\beta AA^*A + A^*A^2 -\gamma (AA^*+A^*A)- \varrho A^*\rbrack = 0, (ii) \lbrack A^*,A^{*2}A-\beta A^*AA^* + AA^{*2} -\gamma^* (A^*A+AA^*)- \varrho^* A\rbrack = 0, where $\lbrack r,s\rbrack $ means $rs-sr$. We call these relations the Tridiagonal relations. For $\beta = q+q^{-1}$, $\gamma = \gamma^*=0$, $\varrho=\varrho^*=0$, the Tridiagonal relations are the $q$-Serre relations. For $\beta = 2$, $\gamma = \gamma^*=0$, $\varrho=b^2$, $\varrho^*=b^{*2}$, the Tridiagonal relations are the Dolan-Grady relations. In the first part of this paper, we survey what is known about irreducible finite dimensional $T$-modules. We focus on how these modules are related to the Leonard pairs recently introduced by the present author, and the more general Tridiagonal pairs recently introduced by Ito, Tanabe, and the present author. In the second part of the paper, we construct an infinite dimensional irreducible $T$-module based on the Askey-Wilson polynomials.