A bilinear form for tridiagonal pairs of q-Serre type

Let V denote a vector space of finite, positive dimension, and let denote the vector space dual of V. Let A, A∗ be a tridiagonal pair on V, and let , be linear transformations such that for all v∈V and , and . We show that , is a tridiagonal pair on . We then show that if A, A∗ is of q-Serre type, then , is isomorphic to A, A∗. We also show that in this case there exist a unique bilinear form on V such that for all u,v∈V, 〈Au,v〉=〈u,Av〉 and 〈A∗u,v〉=〈u,A∗v〉. Moreover, this bilinear form is nondegenerate and symmetric.