# comodule algebra

Let $H$ be a bialgebra^{}.
A right $H$-comodule algebra is a unital algebra^{} $A$ which is a right $H$-comodule
satisfying

$$t(ab)=t(a)t(b)=\sum {a}_{(1)}{b}_{(1)}\otimes {a}_{(2)}{b}_{(2)},t({1\mathrm{I}}_{A})={1\mathrm{I}}_{A}\otimes {1\mathrm{I}}_{H},$$ | (1) |

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

There is a dual notion of a $H$-module coalgebra.

###### Example 1

Let $H$ be a bialgebra. Then $H$ is itself a $H$-comodule algebra for the right regular coaction $t\mathit{}\mathrm{(}h\mathrm{)}\mathrm{=}\mathrm{\Delta}\mathit{}\mathrm{(}h\mathrm{)}$.

Title | comodule algebra |
---|---|

Canonical name | ComoduleAlgebra |

Date of creation | 2013-03-22 13:26:34 |

Last modified on | 2013-03-22 13:26:34 |

Owner | mhale (572) |

Last modified by | mhale (572) |

Numerical id | 8 |

Author | mhale (572) |

Entry type | Definition |

Classification | msc 16W30 |

Related topic | ModuleCoalgebra |

Related topic | ModuleAlgebra |

Related topic | ComoduleCoalgebra |