grouplike elements

Let $(C,\mathrm{\Delta },\epsilon )$ be a coalgebra over a field $k$.

Definition. The element $g\in C$ is called grouplike iff $g\ne 0$ and $\mathrm{\Delta }(g)=g\otimes g$. 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 bialgebra, then $G(C)\ne \mathrm{\varnothing }$. In particular Hopf algebras always have grouplike elements.

$1)$ If $g\in G(C)$, then it follows from the counit property that $\epsilon (g)=1$.

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

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