A bilinear form for tridiagonal pairs of q-Serre type Original Research Article

Let V denote a vector space of finite, positive dimension, and let image denote the vector space dual of V. Let A, A* be a tridiagonal pair on V, and let image, image be linear transformations such that for all vset membership, variantV and image, image and image. We show that image, image is a tridiagonal pair on image. We then show that if A, A* is of q-Serre type, then image, image 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,vset membership, variantV, left angle bracketAu,vright-pointing angle bracket=left angle bracketu,Avright-pointing angle bracket and left angle bracketA*u,vright-pointing angle bracket=left angle bracketu,A*vright-pointing angle bracket. Moreover, this bilinear form is nondegenerate and symmetric.