Representations of finite-dimensional Hopf algebras

Let H denote a finite-dimensional Hopf algebra with antipode S over a field k. We give a new proof of the fact, due to Oberst and Schneider [Manuscripta Math. 8 (1973), 217-241], that H is a symmetric algebra if and only if H is unimodular and S-2 is inner. If H is involuntary and not semisimple, then the dimensions of all projective H-modules are shown to be divisible by char ac. In the case where Ik is a splitting field for H, we give a formula for the rank of the Cartan matrix of H, reduced mod char k, in terms of an integral for H. Explicit computations of the Cartan matrix, the ring structure of G(o)(H), and the structure of the principal indecomposable modules are carried out for certain specific Hopf algebras, in particular for the restricted enveloping algebras of completely solvable p-Lie algebras and of sl(2, k). (C) 1997 Academic Press.