The Drinfel'd Polynomial of a Tridiagonal Pair

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A : V → V and A : V → V that satisfy the following conditions: (i) each of A,A is diagonalizable; (ii) there exists an ordering {Vi} d i=0 of the eigenspaces of A such that A Vi ⊆ Vi−1+Vi+Vi+1 for 0 ≤ i ≤ d, where V−1 = 0 and Vd+1 = 0; (iii) there exists an ordering {V ∗ i } δ i=0 of the eigenspaces of A such that AV ∗ i ⊆ V ∗ i−1 + V ∗ i + V ∗ i+1 for 0 ≤ i ≤ δ, where V ∗ −1 = 0 and V ∗ δ+1 = 0; (iv) there is no subspace W of V such that AW ⊆ W , A W ⊆ W , W 6= 0, W 6= V . We call such a pair a tridiagonal pair on V . It is known that d = δ and for 0 ≤ i ≤ d the dimensions of Vi, Vd−i, V ∗ i , V ∗ d−i coincide. The pair A,A ∗ is called sharp whenever dimV0 = 1. It is known that if F is algebraically closed then A,A ∗ is sharp. Assuming A,A is sharp, we use the data Φ = (A; {Vi} d i=0;A ; {V ∗ i } d i=0) to define a polynomial P in one variable and degree at most d. We show that P remains invariant if Φ is replaced by (A; {Vd−i} d i=0;A ; {V ∗ i } d i=0) or (A; {Vi} d i=0;A ; {V ∗ d−i} d i=0) or (A; {V ∗ i } d i=0;A; {Vi} d i=0). We call P the Drinfel’d polynomial of A,A . We explain how P is related to the classical Drinfel’d polynomial from the theory of Lie algebras and quantum groups. We expect that the roots of P will be useful in a future classification of the sharp tridiagonal pairs. We compute the roots of P for the case in which Vi and V ∗ i have dimension 1 for 0 ≤ i ≤ d.