Representations of Finite Dimensional Hopf Algebras

Let H denote a nite dimensional Hopf algebra with antipode S over a eld |. We give a new proof of the fact, due to Oberst and Schneider OS], that H is a symmetric algebra if and only if H is unimodular and S 2 is inner. If H is involutory and not semisimple, then the dimensions of all projective H-modules are shown to be divisible by char|. In the case where |is a splitting eld for H , we give a formula for the rank of the Cartan matrix of H , reduced mod char| , in terms of an integral for H. Explicit computations of the Cartan matrix, the ring structure of G 0 (H), and the structure of the principal indecomposable modules are carried out for certain speciic Hopf algebras, in particular for the restricted enveloping algebras of completely solvable p-Lie algebras and of sl(2; |).