# 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 |