Cogroups in the category of connected graded algebras whose inverse and antipode coincide

Abstract

Let A be a cogroup in the category of connected graded algebras over a commutative ring R. Let nu denote the inverse of A and chi the antipode of the underlying Hopf algebra of A. We clarify the differences and similarities of nu and chi, and show that nu coincides with chi if and only if A is commutative as a graded algebra. Let A^co_CG be the category of cogroups satisfying these equivalent conditions. If R is a field, the category A^co_CG is completely determined. We also establish an equivalence of the full subcategory of A^co_CG consisting of objects of finite type with a full subcategory of the category of positively graded R-modules without any assumption on R. The results in the case of R=Q are applied to the theory of co-H-groups.