Editing Linear algebraic group (section)

==Definitions==
For an [[algebraically closed field]] ''k'', much of the structure of an [[algebraic variety]] ''X'' over ''k'' is encoded in its set ''X''(''k'') of ''k''-[[rational point]]s, which allows an elementary definition of a linear algebraic group. First, define a function from the abstract group ''GL''(''n'',''k'') to ''k'' to be '''regular''' if it can be written as a polynomial in the entries of an ''n''×''n'' matrix ''A'' and in 1/det(''A''), where det is the [[determinant]]. Then a '''linear algebraic group''' ''G'' over an algebraically closed field ''k'' is a subgroup ''G''(''k'') of the abstract group ''GL''(''n'',''k'') for some natural number ''n'' such that ''G''(''k'') is defined by the vanishing of some set of regular functions.

For an arbitrary field ''k'', algebraic varieties over ''k'' are defined as a special case of [[scheme (mathematics)|schemes]] over ''k''. In that language, a '''linear algebraic group''' ''G'' over a field ''k'' is a [[smooth scheme|smooth]] closed subgroup scheme of ''GL''(''n'') over ''k'' for some natural number ''n''. In particular, ''G'' is defined by the vanishing of some set of [[regular function]]s on ''GL''(''n'') over ''k'', and these functions must have the property that for every commutative ''k''-[[associative algebra|algebra]] ''R'', ''G''(''R'') is a subgroup of the abstract group ''GL''(''n'',''R''). (Thus an algebraic group ''G'' over ''k'' is not just the abstract group ''G''(''k''), but rather the whole family of groups ''G''(''R'') for commutative ''k''-algebras ''R''; this is the philosophy of describing a scheme by its [[functor of points]].)

In either language, one has the notion of a '''[[group homomorphism|homomorphism]]''' of linear algebraic groups. For example, when ''k'' is algebraically closed, a homomorphism from ''G'' ⊂ ''GL''(''m'') to ''H'' ⊂ ''GL''(''n'') is a homomorphism of abstract groups ''G''(''k'') → ''H''(''k'') which is defined by regular functions on ''G''. This makes the linear algebraic groups over ''k'' into a [[category (mathematics)|category]]. In particular, this defines what it means for two linear algebraic groups to be [[category (mathematics)#Types of morphisms|isomorphic]].

In the language of schemes, a linear algebraic group ''G'' over a field ''k'' is in particular a '''[[group scheme]]''' over ''k'', meaning a scheme over ''k'' together with a ''k''-point 1 ∈ ''G''(''k'') and morphisms 
:<math>m\colon G \times_k G \to G, \;  i\colon G \to G</math>
over ''k'' which satisfy the usual axioms for the multiplication and inverse maps in a group (associativity, identity, inverses). A linear algebraic group is also smooth and of [[Glossary of scheme theory#finite type (locally)|finite type]] over ''k'', and it is [[affine scheme|affine]] (as a scheme). Conversely, every affine group scheme ''G'' of finite type over a field ''k'' has a [[faithful representation]] into ''GL''(''n'') over ''k'' for some ''n''.<ref>Milne (2017), Corollary 4.10.</ref> An example is the embedding of the additive group ''G''<sub>''a''</sub> into ''GL''(2), as mentioned above. As a result, one can think of linear algebraic groups either as matrix groups or, more abstractly, as smooth affine group schemes over a field. (Some authors use "linear algebraic group" to mean any affine group scheme of finite type over a field.)

For a full understanding of linear algebraic groups, one has to consider more general (non-smooth) group schemes. For example, let ''k'' be an algebraically closed field of [[characteristic (algebra)|characteristic]] ''p'' > 0. Then the homomorphism ''f'': ''G''<sub>''m''</sub> → ''G''<sub>''m''</sub> defined by ''x'' ↦ ''x''<sup>''p''</sup> induces an isomorphism of abstract groups ''k''* → ''k''*, but ''f'' is not an isomorphism of algebraic groups (because ''x''<sup>1/''p''</sup> is not a regular function). In the language of group schemes, there is a clearer reason why ''f'' is not an isomorphism: ''f'' is surjective, but it has nontrivial [[kernel (algebra)|kernel]], namely the [[multiplicative group#Group scheme of roots of unity|group scheme μ<sub>''p''</sub>]] of ''p''th roots of unity. This issue does not arise in characteristic zero. Indeed, every group scheme of finite type over a field ''k'' of characteristic zero is smooth over ''k''.<ref>Milne (2017), Corollary 8.39.</ref> A group scheme of finite type over any field ''k'' is smooth over ''k'' if and only if it is '''geometrically reduced''', meaning that the [[fiber product of schemes|base change]] <math>G_{\overline k}</math> is [[reduced scheme|reduced]], where <math>\overline k</math> is an [[algebraic closure]] of ''k''.<ref>Milne (2017), Proposition 1.26(b).</ref>

Since an affine scheme ''X'' is determined by its [[ring (mathematics)|ring]] ''O''(''X'') of regular functions, an affine group scheme ''G'' over a field ''k'' is determined by the ring ''O''(''G'') with its structure of a [[Hopf algebra]] (coming from the multiplication and inverse maps on ''G''). This gives an [[equivalence of categories]] (reversing arrows) between affine group schemes over ''k'' and commutative Hopf algebras over ''k''. For example, the Hopf algebra corresponding to the multiplicative group ''G''<sub>''m''</sub> = ''GL''(1) is the [[Laurent polynomial]] ring ''k''[''x'', ''x''<sup>−1</sup>], with comultiplication given by
:<math>x \mapsto x \otimes x.</math>

===Basic notions===
For a linear algebraic group ''G'' over a field ''k'', the [[identity component]] ''G''<sup>o</sup> (the [[connected component (topology)|connected component]] containing the point 1) is a [[normal subgroup]] of finite [[index of a subgroup|index]]. So there is a [[group extension]]
:<math>1 \to G^\circ \to G \to F \to 1, </math>
where ''F'' is a finite algebraic group. (For ''k'' algebraically closed, ''F'' can be identified with an abstract finite group.) Because of this, the study of algebraic groups mostly focuses on connected groups.

Various notions from [[group theory|abstract group theory]] can be extended to linear algebraic groups. It is straightforward to define what it means for a linear algebraic group to be [[abelian group|commutative]], [[nilpotent group|nilpotent]], or [[solvable group|solvable]], by analogy with the definitions in abstract group theory. For example, a linear algebraic group is '''solvable''' if it has a [[composition series]] of linear algebraic subgroups such that the quotient groups are commutative. Also, the [[normalizer]], the [[center of a group|center]], and the [[centralizer]] of a closed subgroup ''H'' of a linear algebraic group ''G'' are naturally viewed as closed subgroup schemes of ''G''. If they are smooth over ''k'', then they are linear algebraic groups as defined above.

One may ask to what extent the properties of a connected linear algebraic group ''G'' over a field ''k'' are determined by the abstract group ''G''(''k''). A useful result in this direction is that if the field ''k'' is [[perfect field|perfect]] (for example, of characteristic zero), ''or'' if ''G'' is reductive (as defined below), then ''G'' is [[unirational]] over ''k''. Therefore, if in addition ''k'' is infinite, the group ''G''(''k'') is [[Zariski dense]] in ''G''.<ref>Borel (1991), Theorem 18.2 and Corollary 18.4.</ref> For example, under the assumptions mentioned, ''G'' is commutative, nilpotent, or solvable if and only if ''G''(''k'') has the corresponding property.

The assumption of connectedness cannot be omitted in these results. For example, let ''G'' be the group μ<sub>''3''</sub> ⊂ ''GL''(1) of cube roots of unity over the [[rational number]]s '''Q'''. Then ''G'' is a linear algebraic group over '''Q''' for which ''G''('''Q''') = 1 is not Zariski dense in ''G'', because <math>G(\overline {\mathbf Q})</math> is a group of order 3.

Over an algebraically closed field, there is a stronger result about algebraic groups as algebraic varieties: every connected linear algebraic group over an algebraically closed field is a [[rational variety]].<ref>Borel (1991), Remark 14.14.</ref>