# subcoalgebras and coideals

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

Definition. Vector subspace $D\subseteq C$ is called subcoalgebra iff $\mathrm{\Delta}(D)\subseteq D\otimes D$.

Definition. Vector subspace $I\subseteq C$ is is called coideal iff $\mathrm{\Delta}(I)\subseteq I\otimes C+C\otimes I$ and $\epsilon (I)=0$.

One can show that if $D\subseteq C$ is a subcoalgebra, then $(D,{\mathrm{\Delta}}_{|D},{\epsilon}_{|D})$ is also a coalgebra. On the other hand, if $I\subseteq C$ is a coideal, then we can cannoicaly introduce a coalgebra structure^{} on the quotient space^{} $C/I$. More precisely, if $x\in C$ and $\mathrm{\Delta}(x)=\sum {a}_{i}\otimes {b}_{i}$, then we define

$${\mathrm{\Delta}}^{\prime}:C/I\to (C/I)\otimes (C/I);$$ |

$${\mathrm{\Delta}}^{\prime}(x+I)=\sum ({a}_{i}+I)\otimes ({b}_{i}+I)$$ |

and ${\epsilon}^{\prime}:C/I\to k$ as ${\epsilon}^{\prime}(x+I)=\epsilon (x)$. One can show that these two maps are well defined and $(C/I,{\mathrm{\Delta}}^{\prime},{\epsilon}^{\prime})$ is a coalgebra.

Title | subcoalgebras and coideals |
---|---|

Canonical name | SubcoalgebrasAndCoideals |

Date of creation | 2013-03-22 18:49:19 |

Last modified on | 2013-03-22 18:49:19 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 4 |

Author | joking (16130) |

Entry type | Definition |

Classification | msc 16W30 |