The submodule structure of Weyl modules for SL has been determined 3 w x ลฝ . w x in DS, I, K and by Cline unpublished . The main technique used in DS is an analysis of the intertwining homomorphisms between certain Weyl w x modules. The approach in I, K is essentially a direct calculation. Also, Jantzen's translation principle plays an important role in these works. For a few special cases, Irving gives some maximal and primitive elements in ลฝ w x. Verma modules of the Frobenius kernel and Weyl modules see I . In this paper we will find out all maximal and primitive elements in Verma modules of the quantized Frobenius kernel and Weyl modules for type A ; 2 see Sections 2แ3. Thus we understand the structure of the Verma modules and the Weyl modules more explicitly. Our method is based on the main w x w x results in X1 and some ideas in I and is rather simple. In this paper we ลฝ only work with quantized enveloping algebras at roots of 1 Lusztig . version ; for hyperalgebras the approach is completely similar. The method ลฝ w x. also works for type B see X2 . The author also did some computations 2 for type A using the method; although the size of the computation is 3 large, it still does not seem difficult to work out the structure of Weyl modules for type A . 3
1. MAXIMAL AND PRIMITIVE ELEMENTS
In this section we fix notation and give the definition for maximal and w x primitive elements. We refer to L1แL4, X1 for additional information.