Tridiagonal pairs of $q$-Serre type and their linear perturbations

Abstract

A tridiagonal pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on the eigenspaces of the other in a block-tridiagonal fashion. We consider a tridiagonal pair $(A,A^{\ast })$ of $q$-Serre type; for such a pair the maps $A$ and $A^{\ast }$ satisfy the $q$-Serre relations. There is a linear map $K$ in the literature that is used to describe how $A$ and $A^{\ast }$ are related. We investigate a pair of linear maps $B=A$ and $B^{\ast }=t{A}^{\ast }+(1-t)K$, where $t$ is any scalar. Our goal is to find a necessary and sufficient condition on $t$ for the pair $(B,B^{\ast })$ to be a tridiagonal pair. We show that $(B,B^{\ast })$ is a tridiagonal pair if and only if $t\ne 0$ and $P(t(q-q^{-1}{)}^{-2})\ne 0$, where $P$ is a certain polynomial attached to $(A,A^{\ast })$ called the Drinfel'd polynomial.