Linear algebraic group: Difference between revisions

Revision as of 02:50, 26 February 2011

Definition

A linear algebraic group over a field $k$ is defined as an abstract group $G$ , along with an embedding of $G$ as a closed subgroup of the general linear group $GL(n,k)$ for some choice of $n$ .

Note that such an embedding automatically turns $G$ into an algebraic group, because $GL(n,k)$ comes naturally equipped with an algebraic group structure, and any closed subgroup of algebraic group inherits algebraic group structure.

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 ==Definition==
-===Definition with symbols===
+A '''linear algebraic group''' over a [[field]] <math>k</math> is defined as an abstract group <math>G</math>, along with an embedding of <math>G</math> as a [[defining ingredient::closed subgroup of algebraic group|closed subgroup]] of the [[general linear group over a field|general linear group]] <math>GL(n,k)</math> for some choice of <math>n</math>.
-A '''linear algebraic group''' over a [[field]] <math>k</math> is defined as an abstract group <math>G</math>, along with an embedding of <math>G</math> as a closed subgroup of <math>GL(n,k)</math> for some choice of <math>n</math>. Note that such an embedding automatically turns <math>G</math> into an algebraic group, because any closed subgroup of an algebraic group automatically inherits the structure of an algebraic group.
+Note that such an embedding automatically turns <math>G</math> into an algebraic group, because <math>GL(n,k)</math> comes naturally equipped with an [[defining ingredient::algebraic group]] structure, and any [[closed subgroup of algebraic group inherits algebraic group structure]].