The structure of a tridiagonal pair Original Research Article

Let image denote a field and let V denote a vector space over image with finite positive dimension.We consider a pair of image-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 image of the eigenspaces of A such that A*Visubset of or equal toVi-1+Vi+Vi+1 for 0less-than-or-equals, slantiless-than-or-equals, slantd, where V-1=0 and Vd+1=0; (iii) there exists an ordering image of the eigenspaces of A* such that image for 0less-than-or-equals, slantiless-than-or-equals, slantδ, where image and image; (iv) there is no subspace W of V such that AWsubset of or equal toW,A*Wsubset of or equal toW,W≠0,W≠V.We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0less-than-or-equals, slantiless-than-or-equals, slantd the dimensions of image coincide.In this paper we show that the following (i)–(iv) hold provided that image is algebraically closed: (i) Each of image has dimension 1.(ii) There exists a nondegenerate symmetric bilinear form left angle bracket,right-pointing angle bracket on V such that 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 for all u,vset membership, variantV.(iii) There exists a unique anti-automorphism of End(V) that fixes each of A,A*.(iv) The pair A,A* is determined up to isomorphism by the data image, where θi (resp.image) is the eigenvalue of A (resp.A*) on Vi (resp.image), andimage is the split sequence of A,A* corresponding to image and image.