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⁎) of q-Serre type; for such a pair the maps A and A⁎ satisfy the q-Serre relations. There is a linear map K in the literature that is used to describe how A and A⁎ are related. We investigate a pair of linear maps B=A and B⁎=tA⁎+(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⁎) to be a tridiagonal pair. We show that (B,B⁎) is a tridiagonal pair if and only if t≠0 and P(t(q−q−1)−2)≠0, where P is a certain polynomial attached to (A,A⁎) called the Drinfel'd polynomial.