Linear algebraic group: Difference between revisions
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===In terms of embedding into general linear group=== | ===In terms of embedding into general linear group=== | ||
A '''linear algebraic group''' over a [[field]] <math>k</math> is defined as an [[algebraic group]] <math>G</math> such that there exists 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> (and this embedding is a morphism of algebraic varieties). | A '''linear algebraic group''' or '''affine algebraic group''' over a [[field]] <math>k</math> is defined as an [[algebraic group]] <math>G</math> such that there exists 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> (and this embedding is a morphism of algebraic varieties). | ||
Note that such an embedding as a closed subgroup of <math>GL(n,k)</math> automatically gives an algebraic group structure, so if we provide an embedding, we do not need to specify an algebraic group structure separately. This is 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]]. | Note that such an embedding as a closed subgroup of <math>GL(n,k)</math> automatically gives an algebraic group structure, so if we provide an embedding, we do not need to specify an algebraic group structure separately. This is 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]]. | ||
Latest revision as of 17:17, 1 January 2012
Definition
In terms of underlying variety
A linear algebraic group or affine algebraic group is an algebraic group where the underlying algebraic variety is an affine variety.
By a basic theorem of algebraic geometry, any affine algebraic geometry has a faithful linear representation, and can hence be realized as a linear algebraic group. Thus, we often view affine algebraic group and linear algebraic group as synonyms.
In terms of embedding into general linear group
A linear algebraic group or affine algebraic group over a field is defined as an algebraic group such that there exists an embedding of as a closed subgroup of the general linear group for some choice of (and this embedding is a morphism of algebraic varieties).
Note that such an embedding as a closed subgroup of automatically gives an algebraic group structure, so if we provide an embedding, we do not need to specify an algebraic group structure separately. This is because comes naturally equipped with an algebraic group structure, and any closed subgroup of algebraic group inherits algebraic group structure.