Three variations on the linear independence of grouplikes in a coalgebra.

Trois variations sur l'indépendance linéaire des éléments de type groupe dans une cogèbre.

Gérard Henry Edmond Duchamp (1) , Darij Grinberg (2) , Vincel Ngoc Hoang Minh (3)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 Drexel University
3 Université de Lille

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.

Domaines: Combinatoire [math.CO]; Algèbres quantiques [math.QA]

Gérard Henry Edmond Duchamp, Darij Grinberg, Vincel Ngoc Hoang Minh. Three variations on the linear independence of grouplikes in a coalgebra.. 2021. ⟨hal-02943601v5⟩