On Finite-dimensional Semisimple and Cosemisimple Hopf Algebras in Positive Characteristic

Recently, important progress has been made in the study of finite-dimensional semisimple Hopf algebras over a field of characteristic zero [Mo and references therein]. Yet, very little is known over a field of positive characteristic. In this paper we prove some results on finite-dimensional semisimple and cosemisimple Hopf algebras A over a field of positive characteristic, notably Kaplansky’s 5th conjecture on the order of the antipode of A [K]. These results have already been proved over a field of characteristic zero, so in a sense we demonstrate that it is sufficient to consider semisimple Hopf algebras over such a field (they are also cosemisimple [LR2]), and then to use our Lifting Theorem 2.1 to prove it for semisimple and cosemisimple Hopf algebras over a field of positive characteristic. In our proof of Lifting Theorem 2.1 we use standard arguments of deformation theory from positive to zero characteristic. The key ingredient of the proof is the theorem that the bialgebra cohomology groups of A vanish [St]. We conclude the paper by proving a weak form of Kaplansky’s 6th conjecture for Hopf algebras which are defined over Z.