On FiniteDimensional Semisimple and Cosemisimple Hopf Algebras in Positive Characteristic
Abstract
Recently, important progress has been made in the study of finitedimensional semisimple Hopf algebras over a field of characteristic zero. Yet, very little is known over a field of positive characteristic. In this paper we prove some results on finitedimensional 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. 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), 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 1998
 DOI:
 10.48550/arXiv.math/9805106
 arXiv:
 arXiv:math/9805106
 Bibcode:
 1998math......5106E
 Keywords:

 Mathematics  Quantum Algebra
 EPrint:
 11 pages, amstex