Coactions on Spaces of Morphisms

Abstract

We study certain comodule structures on spaces of linear morphisms between H-comodules, where H is a Hopf algebra over the field k. We apply the results to show that H has non-zero integrals if and only if there exists a non-zero finite dimensional injective right H-comodule. Using this approach, we prove an extension of a result of Sullivan, by showing that if H is involutory and has non-zero integrals, and there exists an injective indecomposable right comodule whose dimension is not a multiple of char(k), then H is cosemisimple. Also we prove without using character theory that if H is cosemisimple and M is an absolutely irreducible right H-comodule, then char(k) does not divide dim(M).

On two finiteness conditions for Hopf algebras with nonzero integral

“…Given M ∈ M H , its injective hull is denoted by E (M ). The next characterization summarizes several results along the years, see [L,Theorems 3 and 10], [DN,Proposition 2.3], [Do1, Lemma 1], [Do2,page 223] and [AC,Theorems 2.3 and 2.8].…”