# affine algebraic group

An affine algebraic group over a field $k$ is quasi-affine variety $G$ (a locally closed subset of affine space) over $k$, which is a equipped with a group such that the multiplication map $m:G\times G\to G$ and inverse map $i:G\to G$ are algebraic^{}.

For example, $k$ is an affine algebraic group over itself with the group law being addition, and as is ${k}^{*}=k-\{0\}$ with the group law multiplication. Other common examples of affine algebraic groups are ${\mathrm{GL}}_{n}k$, the general linear group^{} over $k$ (identifying matrices with affine space) and any algebraic torus over $k$.

Title | affine algebraic group |
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Canonical name | AffineAlgebraicGroup |

Date of creation | 2013-03-22 13:53:33 |

Last modified on | 2013-03-22 13:53:33 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 7 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 14L17 |

Related topic | GroupVariety |