Structure of U _∈ (g)

Abstract

Throughout this chapter we’ll assume that g is a finite dimensional semisimple complex Lie algebra and that e is a primitive £th root of unity in the fieldkof characteristic0,where

P > 3 is odd, and prime to 3 if g contains a factor of type G2. (1)

Our aim in this chapter is to examine how the structure of U_E(5(2(k)) which we found in Chapter III.2 generalises to arbitrary semisimple g. We’ll see (111.6.2) that (Mg) is a PI Hopf triple in the language of Chapter 111.4. Moreover the construction of (I11.5.6) can be used to obtain a structure of Poisson group on the Hopf centre Z0 of U _E (g), so that the representation theory of U _E (g) can be studied with the help of (III.5.8).