The diagonal of a pointed coalgebra and incidence-like structure

Abstract

Vector space decompositions of a pointed coalgebra C over a field reflecting properties of its diagonal map are used by Sweedler [11] to classify the coalgebra, by Heyneman and Radford ([6], [10]) to discuss coreflexivity, and by Taft and Wilson (e.g. [13]) to obtain results about Hopf algebras from the underlying coalgebra. However, this type of structure has been classified in full only in specific cases, and the behavior of the diagonal on a general pointed coalgebra is known only to the extent obtained in the above. Using the structure of the first term of the coradical filtration (cf. [10], [11], [13]), Taft and Wilson [13] gave a vector-space decomposition yielding some information about the highest-weight terms in the diagonal (cf. 1.3 below). In addition, the complete structure of the diagonal is known for incidence coalgebras (on a partially-ordered set), and for those coalgebras which are a sum of their pointed irreducible components (PIC's). Subsequently, other papers have treated the structure of coalgebras from other points of view In this paper, the result of Taft and Wilson is refined to obtain a generalization of the PIC case (2.4 and 2.6), which agrees with the natural structure in the case of incidence coalgebras. Further refinements (3.1 and 3.6) are obtained by the use of certain invariants of the coalgebra, yielding a characterization of PIC coalgebras (3.2).