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 [Formula: see text] 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 [Formula: see text] of the eigenspaces of A* such that [Formula: see text] for 0 ≤ i ≤ δ, where [Formula: see text] and [Formula: see text]; (iv) there is no subspace W of V such that AW ⊆ W, A* W ⊆ W, W ≠ 0, W ≠ 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, [Formula: see text], Vd-i, [Formula: see text] coincide. Denote this common dimension by ρi and call A, A*sharp whenever ρ0 = 1. Let T denote the 𝔽-subalgebra of End 𝔽(V) generated by A, A*. We show: (i) the center Z(T) is a field whose dimension over 𝔽 is ρ0; (ii) the field Z(T) is isomorphic to each of E0TE0, EdTEd, [Formula: see text], [Formula: see text], where Ei (resp. [Formula: see text]) is the primitive idempotent of A (resp. A*) associated with Vi (resp. [Formula: see text]); (iii) with respect to the Z(T)-vector space V the pair A, A* is a sharp tridiagonal pair.