The corepresentation ring of a pointed Hopf algebra of rank one

Abstract

Let [Formula: see text] be an arbitrary pointed Hopf algebra of rank one and [Formula: see text] the group of group-like elements of [Formula: see text]. In this paper, we give the decomposition of a tensor product of finite dimensional indecomposable right [Formula: see text]-comodules into a direct sum of indecomposables. This enables us to describe the corepresentation ring of [Formula: see text] in terms of generators and relations. Such a ring is not commutative if [Formula: see text] is not abelian. We describe all nilpotent elements of the corepresentation ring of [Formula: see text] if [Formula: see text] is a finite abelian group or a particular Hamiltonian group. In this case, all nilpotent elements of the corepresentation ring form a principal ideal which is either zero or generated by a nilpotent element of degree 2.