grouplike elements

Let (C,Δ,ε) be a coalgebra over a field k.

Definition. The element gC is called grouplike iff g0 and Δ(g)=gg. The set of all grouplike elements in a coalgebra C is denoted by G(C).

Properties. 0) The set G(C) can be empty, but (for example) if C can be turned into a bialgebraPlanetmathPlanetmathPlanetmath, then G(C). In particular Hopf algebrasPlanetmathPlanetmath always have grouplike elements.

1) If gG(C), then it follows from the counit property that ε(g)=1.

2) It can be shown that the set G(C) is linearly independentMathworldPlanetmath.

Title grouplike elements
Canonical name GrouplikeElements
Date of creation 2013-03-22 18:58:37
Last modified on 2013-03-22 18:58:37
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Definition
Classification msc 16W30