grouplike elements

Let $(C,\\Delta,\\varepsilon)$ be a coalgebra over a field $k$ .

Definition. The element $g\\in C$ is called grouplike iff $g\eq 0$ and $\\Delta(g)=g\\otimes g$ . 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 $G(C)\eq\\emptyset$ . In particular Hopf algebras always have grouplike elements.

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

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

单词	GrouplikeElements
释义	grouplike elements Let $(C,\\Delta,\\varepsilon)$ be a coalgebra over a field $k$ . Definition. The element $g\\in C$ is called grouplike iff $g\eq 0$ and $\\Delta(g)=g\\otimes g$ . 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 $G(C)\eq\\emptyset$ . In particular Hopf algebras always have grouplike elements. $1)$ If $g\\in G(C)$ , then it follows from the counit property that $\\varepsilon(g)=1$ . $2)$ It can be shown that the set $G(C)$ is linearly independent.
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