On a type of differential Hopf algebras

Abstract

i. When we plan to organize a theory of Hopf algebras modelled on mod p K-theory, we will meet with two significant difference from the ordinary theory of Hopf algebras. The first point is: the classical Hopf algebras modelled on ordinary homology and cohomology of H-spaces is non-negatively graded and can be discussed sometimes by making use of an induction argument on degrees (cf. [7] etc.), but our Hopf algebra is Z2-graded, so we can not use such arguments. We solve some of these type of problems by using the two filtrations (F-filtration by multiplication and G-filtration by comultiplication) originally due to Browder [4]. The second point is: in the classical theory of Hopf algebras the relation