Abstract :
LetHdenote a finite-dimensional Hopf algebra with antipodeSover a field image. We give a new proof of the fact, due to[OS], thatHis a symmetric algebra if and only ifHis unimodular andS2is inner. IfHis involutory and not semisimple, then the dimensions of all projectiveH-modules are shown to be divisible by char image. In the case where image is a splitting field forH, we give a formula for the rank of the Cartan matrix ofH, reduced mod char image, in terms of an integral forH. Explicit computations of the Cartan matrix, the ring structure ofG0(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 solvablep-Lie algebras and ofsl(2, image).
