# module coalgebra

Let $H$ be a bialgebra^{}.
A left $H$-module coalgebra is a coalgebra $A$ which is a left $H$-module
with action $h\u25b7a$ satisfying

$$\mathrm{\Delta}(h\u25b7a)=\sum ({h}_{(1)}\u25b7{a}_{(1)})\otimes ({h}_{(2)}\u25b7{a}_{(2)}),\epsilon (h\u25b7a)=\epsilon (h)\epsilon (a),$$ | (1) |

for all $h\in H$ and $a\in A$.

There is a dual notion of a $H$-comodule algebra.

###### Example 1

Let $H$ be a bialgebra.
Then $H$ is itself a $H$-module coalgebra for the left regular^{} action
$g\mathrm{\u25b7}h\mathrm{=}g\mathit{}h$.

Title | module coalgebra |
---|---|

Canonical name | ModuleCoalgebra |

Date of creation | 2013-03-22 13:26:37 |

Last modified on | 2013-03-22 13:26:37 |

Owner | mhale (572) |

Last modified by | mhale (572) |

Numerical id | 9 |

Author | mhale (572) |

Entry type | Definition |

Classification | msc 16W30 |

Related topic | ComoduleAlgebra |

Related topic | ModuleAlgebra |

Related topic | ComoduleCoalgebra |