Three variations on the linear independence of grouplikes in a coalgebra

TL;DR

This paper explores three variants of the linear independence of grouplike elements in coalgebras, extending the classical result to broader algebraic structures and characters.

Contribution

It introduces three new variants of linear independence for grouplike elements, including generalizations to coalgebras over rings and characters of bialgebras.

Findings

Linear independence of grouplike elements over fields

Weaker independence statements over rings

Linear independence of characters in bialgebras

Abstract

The grouplike elements of a coalgebra over a field are known to be linearly independent over said field. Here we prove three variants of this result. One is a generalization to coalgebras over a commutative ring (in which case the linear independence has to be replaced by a weaker statement). Another is a stronger statement that holds (un-der stronger assumptions) in a commutative bialgebra. The last variant is a linear independence result for characters (as opposed to grouplike elements) of a bialgebra.