Affine transformations of a sharp tridiagonal pair

Let KK denote a field and let V   denote a vector space over KK with finite positive dimension. By a tridiagonal pair  , we mean an ordered pair A,A⁎A,A⁎ of KK-linear transformations from V to V   that satisfy the following conditions: (i) each of A,A⁎A,A⁎ is diagonalizable; (ii) there exists an ordering {Vi}i=0d of the eigenspaces of A   such that A⁎Vi⊆Vi−1+Vi+Vi+1A⁎Vi⊆Vi−1+Vi+Vi+1 (0≤i≤d0≤i≤d), where V−1=0V−1=0, Vd+1=0Vd+1=0; (iii) there exists an ordering {Vi⁎}i=0δ of the eigenspaces of A⁎A⁎ such that AVi⁎⊆Vi−1⁎+Vi⁎+Vi+1⁎ (0≤i≤δ0≤i≤δ), where V−1⁎=0, Vδ+1⁎=0; (iv) there is no subspace W of V   such that AW⊆WAW⊆W, A⁎W⊆WA⁎W⊆W, W≠0W≠0, W≠VW≠V. It is known that ηA+μIηA+μI, η⁎A⁎+μ⁎Iη⁎A⁎+μ⁎I is also a tridiagonal pair on V  , where η,μ,η⁎,μ⁎η,μ,η⁎,μ⁎ are scalars in KK with η,η⁎η,η⁎ nonzero. In this paper we give the necessary and sufficient conditions for these tridiagonal pairs to be isomorphic to A,A⁎A,A⁎ or A⁎,AA⁎,A. We do this under a mild assumption, called the sharp condition.