Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra

Then U Q is known to be the enveloping algebra of the simple Lie algebra g over Q corresponding to (a).Chevalley [2] has proved that any U Q-module of finite dimension over Q admits a lattice which is stable under the subring U of U Q generated by the -(N) -N ....,.{N)-N Ei = Ei IN! and Fi = Fi IN! (1 ~ i ~ n, N ;::: 0), and Kostant [7] constructed a nice Z-basis for U. Then u can be defined as the subring of U ® Fp = U F generated by the p elements E~ I) and Pi 1 ) (1::; i ~ n).The pth powers of these generators are zero and in fact u is of finite dimension (= pdim9) over Fp'