Difference between revisions of "Linear algebraic group"

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==Definition==
 
==Definition==
  
===Definition with symbols===
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===In terms of underlying variety===
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.
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A '''linear algebraic group''' or '''affine algebraic group''' is an [[algebraic group]] where the underlying [[algebraic variety]] is an [[affine variety]].
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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.
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===In terms of embedding into general linear group===
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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).  
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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 k is defined as an algebraic group G such that there exists an embedding of G as a closed subgroup of the general linear group GL(n,k) for some choice of n (and this embedding is a morphism of algebraic varieties).

Note that such an embedding as a closed subgroup of GL(n,k) 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 GL(n,k) comes naturally equipped with an algebraic group structure, and any closed subgroup of algebraic group inherits algebraic group structure.