Editing Linear algebraic group

{{Short description|Subgroup of the group of invertible n×n matrices}}
{{Group theory sidebar|Algebraic}}

In [[mathematics]], a '''linear algebraic group'''  is a [[subgroup]] of the [[group (mathematics)|group]] of [[invertible matrix|invertible]] <math>n\times n</math> [[Matrix (mathematics)|matrices]] (under [[matrix multiplication]]) that is defined by [[polynomial]] equations. An example is the [[orthogonal group]], defined by the relation <math>M^TM = I_n</math> where <math>M^T</math> is the [[transpose]] of <math>M</math>.

Many [[Lie group]]s can be viewed as linear algebraic groups over the [[Field (mathematics)|field]] of [[Real number|real]] or [[Complex number|complex]] numbers.  (For example, every [[compact Lie group]] can be regarded as a linear algebraic group over '''R''' (necessarily '''R'''-anisotropic and reductive), as can many noncompact groups such as the [[simple Lie group]] [[special linear group|SL(''n'','''R''')]].) The simple Lie groups were classified by [[Wilhelm Killing]] and [[Élie Cartan]] in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include [[Ludwig Maurer|Maurer]], [[Claude Chevalley|Chevalley]], and {{harvs|txt|last=Kolchin|author1-link=Ellis Kolchin|year=1948}}. In the 1950s, [[Armand Borel]] constructed much of the theory of algebraic groups as it exists today.

One of the first uses for the theory was to define the [[Chevalley group]]s.

==Examples==
For a [[positive integer]] <math>n</math>, the [[general linear group]] <math>GL(n)</math> over a field <math>k</math>, consisting of all invertible <math>n\times n</math> matrices, is a linear algebraic group over <math>k</math>. It contains the subgroups 
:<math>U \subset B \subset GL(n)</math>
consisting of matrices of the form, resp.,
:<math>\left ( \begin{array}{cccc} 1 & * & \dots & * \\ 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & * \\ 0 & \dots & 0 & 1\end{array} \right )</math> and <math>\left ( \begin{array}{cccc} * & * & \dots & * \\ 0 & * & \ddots & \vdots \\ \vdots & \ddots & \ddots & * \\ 0 & \dots & 0 & *\end{array} \right )</math>.
The group <math>U</math> is an example of a [[unipotent]] linear algebraic group, the group <math>B</math> is an example of a [[Solvable group|solvable]] algebraic group called the [[Borel subgroup]] of <math>GL(n)</math>. It is a consequence of the [[Lie-Kolchin theorem]] that any connected solvable subgroup of <math>\mathrm{GL}(n)</math> is conjugated into <math>B</math>. Any unipotent subgroup can be conjugated into <math>U</math>.

Another algebraic subgroup of <math>\mathrm{GL}(n)</math> is the [[special linear group]] <math>\mathrm{SL}(n)</math> of matrices with determinant 1.

The group <math>\mathrm{GL}(1)</math> is called the '''[[multiplicative group]]''', usually denoted by <math> \mathbf G_{\mathrm m}</math>. The group of <math>k</math>-points <math>\mathbf G_{\mathrm m}(k)</math> is the multiplicative group <math>k^*</math> of nonzero elements of the field <math>k</math>. The '''additive group''' <math>\mathbf G_{\mathrm a}</math>, whose <math>k</math>-points are isomorphic to the additive group of <math>k</math>, can also be expressed as a matrix group, for example as the subgroup <math>U</math> in <math>\mathrm{GL}(2)</math> : 
:<math>\begin{pmatrix}
 1 & * \\
 0 & 1
\end{pmatrix}.</math>

These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of their [[linear representation]]s (as algebraic groups). Every representation of the multiplicative group <math>\mathbf G_{\mathrm m}</math> is a [[direct sum]] of [[irreducible representation]]s. (Its irreducible representations all have dimension 1, of the form <math>x \mapsto x^n</math> for an integer <math>n</math>.) By contrast, the only irreducible representation of the additive group <math>\mathbf G_{\mathrm a}</math> is the trivial representation. So every representation of <math>\mathbf G_{\mathrm a}</math> (such as the 2-dimensional representation above) is an iterated [[composition series|extension]] of trivial representations, not a direct sum (unless the representation is trivial). The structure theory of linear algebraic groups analyzes any linear algebraic group in terms of these two basic groups and their generalizations, tori and unipotent groups, as discussed below.

==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>

==The Lie algebra of an algebraic group==
The [[Lie algebra]] <math>\mathfrak g</math> of an algebraic group ''G'' can be defined in several equivalent ways: as the [[Zariski tangent space|tangent space]] ''T''<sub>1</sub>(''G'') at the identity element 1 ∈ ''G''(''k''), or as the space of left-invariant [[derivation (abstract algebra)|derivation]]s. If ''k'' is algebraically closed, a derivation ''D'': ''O''(''G'') → ''O''(''G'') over ''k'' of the coordinate ring of ''G'' is '''left-invariant''' if
:<math>D \lambda_x = \lambda_x D</math>
for every ''x'' in ''G''(''k''), where λ<sub>''x''</sub>: ''O''(''G'') → ''O''(''G'') is induced by left multiplication by ''x''. For an arbitrary field ''k'', left invariance of a derivation is defined as an analogous equality of two linear maps ''O''(''G'') → ''O''(''G'') ⊗''O''(''G'').<ref>Milne (2017), section 10.e.</ref> The Lie bracket of two derivations is defined by [''D''<sub>1</sub>, ''D''<sub>2</sub>] =''D''<sub>1</sub>''D''<sub>2</sub> − ''D''<sub>2</sub>''D''<sub>1</sub>.

The passage from ''G'' to <math>\mathfrak g</math> is thus a process of [[differentiation (mathematics)|differentiation]]. For an element ''x'' ∈ ''G''(''k''), the derivative at 1 ∈ ''G''(''k'') of the [[conjugation (group theory)|conjugation]] map ''G'' → ''G'', ''g'' ↦ ''xgx''<sup>−1</sup>, is an [[automorphism]] of <math>\mathfrak g</math>, giving the [[adjoint representation]]:
:<math>\operatorname{Ad}\colon G \to \operatorname{Aut}(\mathfrak g).</math>

Over a field of characteristic zero, a connected subgroup ''H'' of a linear algebraic group ''G'' is uniquely determined by its Lie algebra <math>\mathfrak h \subset \mathfrak g</math>.<ref>Borel (1991), section 7.1.</ref> But not every Lie subalgebra of <math>\mathfrak g</math> corresponds to an algebraic subgroup of ''G'', as one sees in the example of the torus ''G'' = (''G''<sub>''m''</sub>)<sup>2</sup> over '''C'''. In positive characteristic, there can be many different connected subgroups of a group ''G'' with the same Lie algebra (again, the torus ''G'' = (''G''<sub>''m''</sub>)<sup>2</sup> provides examples). For these reasons, although the Lie algebra of an algebraic group is important, the structure theory of algebraic groups requires more global tools.

==Semisimple and unipotent elements==
{{main|Jordan–Chevalley decomposition}}
For an algebraically closed field ''k'', a matrix ''g'' in ''GL''(''n'',''k'') is called '''semisimple''' if it is [[diagonalizable]], and '''[[unipotent]]''' if the matrix ''g'' − 1 is [[nilpotent]]. Equivalently, ''g'' is unipotent if all [[eigenvalue]]s of ''g'' are equal to 1. The [[Jordan canonical form]] for matrices implies that every element ''g'' of ''GL''(''n'',''k'') can be written uniquely as a product ''g'' = ''g''<sub>ss</sub>''g''<sub>u</sub> such that ''g''<sub>ss</sub> is semisimple, ''g''<sub>u</sub> is unipotent, and ''g''<sub>''ss''</sub> and ''g''<sub>u</sub> [[commuting matrices|commute]] with each other.

For any field ''k'', an element ''g'' of ''GL''(''n'',''k'') is said to be semisimple if it becomes diagonalizable over the algebraic closure of ''k''. If the field ''k'' is perfect, then the semisimple and unipotent parts of ''g'' also lie in ''GL''(''n'',''k''). Finally, for any linear algebraic group ''G'' ⊂ ''GL''(''n'') over a field ''k'', define a ''k''-point of ''G'' to be semisimple or unipotent if it is semisimple or unipotent in ''GL''(''n'',''k''). (These properties are in fact independent of the choice of a faithful representation of ''G''.) If the field ''k'' is perfect, then the semisimple and unipotent parts of a ''k''-point of ''G'' are automatically in ''G''. That is (the '''Jordan decomposition'''): every element ''g'' of ''G''(''k'') can be written uniquely as a product ''g'' = ''g''<sub>ss</sub>''g''<sub>u</sub> in ''G''(''k'') such that ''g''<sub>ss</sub> is semisimple, ''g''<sub>u</sub> is unipotent, and ''g''<sub>''ss''</sub> and ''g''<sub>u</sub> commute with each other.<ref>Milne (2017), Theorem 9.18.</ref> This reduces the problem of describing the [[conjugacy class]]es in ''G''(''k'') to the semisimple and unipotent cases.

==Tori==
{{main|Algebraic torus}}
A '''torus''' over an algebraically closed field ''k'' means a group isomorphic to (''G''<sub>''m''</sub>)<sup>''n''</sup>, the [[Cartesian product|product]] of ''n'' copies of the multiplicative group over ''k'', for some natural number ''n''. For a linear algebraic group ''G'', a '''maximal torus''' in ''G'' means a torus in ''G'' that is not contained in any bigger torus. For example, the group of diagonal matrices in ''GL''(''n'') over ''k'' is a maximal torus in ''GL''(''n''), isomorphic to (''G''<sub>''m''</sub>)<sup>''n''</sup>. A basic result of the theory is that any two maximal tori in a group ''G'' over an algebraically closed field ''k'' are [[conjugacy class#Conjugacy of subgroups and general subsets|conjugate]] by some element of ''G''(''k'').<ref>Borel (1991), Corollary 11.3.</ref> The '''rank''' of ''G'' means the dimension of any maximal torus.

For an arbitrary field ''k'', a '''torus''' ''T'' over ''k'' means a linear algebraic group over ''k'' whose base change <math>T_{\overline k}</math> to the algebraic closure of ''k'' is isomorphic to (''G''<sub>''m''</sub>)<sup>''n''</sup> over <math>\overline k</math>, for some natural number ''n''. A '''split torus''' over ''k'' means a group isomorphic to (''G''<sub>''m''</sub>)<sup>''n''</sup> over ''k'' for some ''n''. An example of a non-split torus over the real numbers '''R''' is
:<math>T=\{(x,y)\in A^2_{\mathbf{R}}: x^2+y^2=1\},</math>
with group structure given by the formula for multiplying complex numbers ''x''+''iy''. Here ''T'' is a torus of dimension 1 over '''R'''. It is not split, because its group of real points ''T''('''R''') is the [[circle group]], which is not isomorphic even as an abstract group to ''G''<sub>''m''</sub>('''R''') = '''R'''*.

Every point of a torus over a field ''k'' is semisimple. Conversely, if ''G'' is a connected linear algebraic group such that every element of <math>G(\overline k)</math> is semisimple, then ''G'' is a torus.<ref>Milne (2017), Corollary 17.25</ref>

For a linear algebraic group ''G'' over a general field ''k'', one cannot expect all maximal tori in ''G'' over ''k'' to be conjugate by elements of ''G''(''k''). For example, both the multiplicative group ''G''<sub>''m''</sub> and the circle group ''T'' above occur as maximal tori in ''SL''(2) over '''R'''. However, it is always true that any two '''maximal split tori''' in ''G'' over ''k'' (meaning split tori in ''G'' that are not contained in a bigger ''split'' torus) are conjugate by some element of ''G''(''k'').<ref>Springer (1998), Theorem 15.2.6.</ref> As a result, it makes sense to define the ''' ''k''-rank''' or '''split rank''' of a group ''G'' over ''k'' as the dimension of any maximal split torus in ''G'' over ''k''.

For any maximal torus ''T'' in a linear algebraic group ''G'' over a field ''k'', Grothendieck showed that <math>T_{\overline k}</math> is a maximal torus in <math>G_{\overline k}</math>.<ref>Borel (1991), 18.2(i).</ref> It follows that any two maximal tori in ''G'' over a field ''k'' have the same dimension, although they need not be isomorphic.

==Unipotent groups==
Let ''U''<sub>''n''</sub> be the group of upper-triangular matrices in ''GL''(''n'') with diagonal entries equal to 1, over a field ''k''. A group scheme over a field ''k'' (for example, a linear algebraic group) is called '''unipotent''' if it is isomorphic to a closed subgroup scheme of ''U''<sub>''n''</sub> for some ''n''. It is straightforward to check that the group ''U''<sub>''n''</sub> is nilpotent. As a result, every unipotent group scheme is nilpotent.

A linear algebraic group ''G'' over a field ''k'' is unipotent if and only if every element of <math>G(\overline{k})</math> is unipotent.<ref>Milne (2017), Corollary 14.12.</ref>

The group ''B''<sub>''n''</sub> of upper-triangular matrices in ''GL''(''n'') is a [[semidirect product]]
:<math>B_n = T_n \ltimes U_n,</math>
where ''T''<sub>''n''</sub> is the diagonal torus (''G''<sub>''m''</sub>)<sup>''n''</sup>. More generally, every connected solvable linear algebraic group is a semidirect product of a torus with a unipotent group, ''T'' ⋉ ''U''.<ref>Borel (1991), Theorem 10.6.</ref>

A smooth connected unipotent group over a perfect field ''k'' (for example, an algebraically closed field) has a composition series with all quotient groups isomorphic to the additive group ''G''<sub>''a''</sub>.<ref>Borel (1991), Theorem 15.4(iii).</ref>

==Borel subgroups==
The '''[[Borel subgroup]]s''' are important for the structure theory of linear algebraic groups. For a linear algebraic group ''G'' over an algebraically closed field ''k'', a Borel subgroup of ''G'' means a maximal smooth connected solvable subgroup. For example, one Borel subgroup of ''GL''(''n'') is the subgroup ''B'' of [[upper triangular matrix|upper-triangular matrices]] (all entries below the diagonal are zero).

A basic result of the theory is that any two Borel subgroups of a connected group ''G'' over an algebraically closed field ''k'' are conjugate by some element of ''G''(''k'').<ref>Borel (1991), Theorem 11.1.</ref> (A standard proof uses the [[Borel fixed-point theorem]]: for a connected solvable group ''G'' acting on a [[proper variety]] ''X'' over an algebraically closed field ''k'', there is a ''k''-point in ''X'' which is fixed by the action of ''G''.) The conjugacy of Borel subgroups in ''GL''(''n'') amounts to the [[Lie–Kolchin theorem]]: every smooth connected solvable subgroup of ''GL''(''n'') is conjugate to a subgroup of the upper-triangular subgroup in ''GL''(''n'').

For an arbitrary field ''k'', a Borel subgroup ''B'' of ''G'' is defined to be a subgroup over ''k'' such that, over an algebraic closure <math>\overline k</math> of ''k'', <math>B_{\overline k}</math> is a Borel subgroup of <math>G_{\overline k}</math>. Thus ''G'' may or may not have a Borel subgroup over ''k''.

For a closed subgroup scheme ''H'' of ''G'', the [[quotient space (linear algebra)|quotient space]] ''G''/''H'' is a smooth [[quasi-projective]] scheme over ''k''.<ref>Milne (2017), Theorems 7.18 and 8.43.</ref> A smooth subgroup ''P'' of a connected group ''G'' is called '''parabolic''' if ''G''/''P'' is [[projective variety|projective]] over ''k'' (or equivalently, proper over ''k''). An important property of Borel subgroups ''B'' is that ''G''/''B'' is a projective variety, called the '''flag variety''' of ''G''. That is, Borel subgroups are parabolic subgroups. More precisely, for ''k'' algebraically closed, the Borel subgroups are exactly the minimal parabolic subgroups of ''G''; conversely, every subgroup containing a Borel subgroup is parabolic.<ref>Borel (1991), Corollary 11.2.</ref> So one can list all parabolic subgroups of ''G'' (up to conjugation by ''G''(''k'')) by listing all the linear algebraic subgroups of ''G'' that contain a fixed Borel subgroup. For example, the subgroups ''P'' ⊂ ''GL''(3) over ''k'' that contain the Borel subgroup ''B'' of upper-triangular matrices are ''B'' itself, the whole group ''GL''(3), and the intermediate subgroups
:<math>\left \{ \begin{bmatrix}
 * & * & * \\
 0 & * & * \\
0 & * & *
\end{bmatrix} \right \}</math> and <math>\left \{ \begin{bmatrix}
 * & * & * \\
 * & * & * \\
0 & 0 & *
\end{bmatrix} \right \}.</math>
The corresponding '''[[generalized flag variety|projective homogeneous varieties]]''' ''GL''(3)/''P'' are (respectively): the '''flag manifold''' of all chains of linear subspaces
:<math>0\subset V_1\subset V_2\subset A^3_k</math>
with ''V''<sub>''i''</sub> of dimension ''i''; a point; the '''[[projective space]]''' '''P'''<sup>2</sup> of lines (1-dimensional [[linear subspace]]s) in ''A''<sup>3</sup>; and the dual projective space '''P'''<sup>2</sup> of planes in ''A''<sup>3</sup>.

==Semisimple and reductive groups==
{{main|Reductive group}}
A connected linear algebraic group ''G'' over an algebraically closed field is called '''semisimple''' if every smooth connected solvable normal subgroup of ''G'' is trivial. More generally, a connected linear algebraic group ''G'' over an algebraically closed field is called '''[[reductive group|reductive]]''' if every smooth connected unipotent normal subgroup of ''G'' is trivial.<ref>Milne (2017), Definition 6.46.</ref> (Some authors do not require reductive groups to be connected.) A semisimple group is reductive. A group ''G'' over an arbitrary field ''k'' is called semisimple or reductive if <math>G_{\overline k}</math> is semisimple or reductive. For example, the group ''SL''(''n'') of ''n'' × ''n'' matrices with determinant 1 over any field ''k'' is semisimple, whereas a nontrivial torus is reductive but not semisimple. Likewise, ''GL''(''n'') is reductive but not semisimple (because its center ''G''<sub>''m''</sub> is a nontrivial smooth connected solvable normal subgroup).

Every compact connected Lie group has a '''[[complexification (Lie group)|complexification]]''', which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex reductive groups, up to isomorphism.<ref>Bröcker & tom Dieck (1985), section III.8; Conrad (2014), section D.3.</ref>

A linear algebraic group ''G'' over a field ''k'' is called '''simple''' (or ''k''-'''simple''') if it is semisimple, nontrivial, and every smooth connected normal subgroup of ''G'' over ''k'' is trivial or equal to ''G''.<ref>Conrad (2014), after Proposition 5.1.17.</ref> (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integer ''n'' at least 2 and any field ''k'', the group ''SL''(''n'') over ''k'' is simple, and its center is the group scheme μ<sub>''n''</sub> of ''n''th roots of unity.

Every connected linear algebraic group ''G'' over a perfect field ''k'' is (in a unique way) an extension of a reductive group ''R'' by a smooth connected unipotent group ''U'', called the '''unipotent radical''' of ''G'':
:<math>1\to U\to G\to R\to 1.</math>
If ''k'' has characteristic zero, then one has the more precise '''[[Levi decomposition]]''': every connected linear algebraic group ''G'' over ''k'' is a semidirect product <math>R\ltimes U</math> of a reductive group by a unipotent group.<ref>Conrad (2014), Proposition 5.4.1.</ref>

==Classification of reductive groups==
{{main|Reductive group}}
Reductive groups include the most important linear algebraic groups in practice, such as the [[classical group]]s: ''GL''(''n''), ''SL''(''n''), the [[orthogonal group]]s ''SO''(''n'') and the [[symplectic group]]s ''Sp''(2''n''). On the other hand, the definition of reductive groups is quite "negative", and it is not clear that one can expect to say much about them. Remarkably, [[Claude Chevalley]] gave a complete classification of the reductive groups over an algebraically closed field: they are determined by [[root data]].<ref>Springer (1998), 9.6.2 and 10.1.1.</ref> In particular, simple groups over an algebraically closed field ''k'' are classified (up to quotients by finite central subgroup schemes) by their [[Dynkin diagram]]s. It is striking that this classification is independent of the characteristic of ''k''. For example, the [[exceptional Lie group]]s ''G''<sub>2</sub>, ''F''<sub>4</sub>, ''E''<sub>6</sub>, ''E''<sub>7</sub>, and ''E''<sub>8</sub> can be defined in any characteristic (and even as group schemes over '''Z'''). The [[classification of finite simple groups]] says that most finite simple groups arise as the group of ''k''-points of a simple algebraic group over a finite field ''k'', or as minor variants of that construction.

Every reductive group over a field is the quotient by a finite central subgroup scheme of the product of a torus and some simple groups. For example,
:<math>GL(n)\cong (G_m\times SL(n))/\mu_n.</math>

For an arbitrary field ''k'', a reductive group ''G'' is called '''split''' if it contains a split maximal torus over ''k'' (that is, a split torus in ''G'' which remains maximal over an algebraic closure of ''k''). For example, ''GL''(''n'') is a split reductive group over any field ''k''. Chevalley showed that the classification of ''split'' reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. For example, every nondegenerate [[quadratic form]] ''q'' over a field ''k'' determines a reductive group ''SO''(''q''), and every [[central simple algebra]] ''A'' over ''k'' determines a reductive group ''SL''<sub>1</sub>(''A''). As a result, the problem of classifying reductive groups over ''k'' essentially includes the problem of classifying all quadratic forms over ''k'' or all central simple algebras over ''k''. These problems are easy for ''k'' algebraically closed, and they are understood for some other fields such as [[number field]]s, but for arbitrary fields there are many open questions.

==Applications==
===Representation theory===
One reason for the importance of reductive groups comes from representation theory. Every irreducible representation of a unipotent group is trivial. More generally, for any linear algebraic group ''G'' written as an extension
:<math>1\to U\to G\to R\to 1</math>
with ''U'' unipotent and ''R'' reductive, every irreducible representation of ''G'' factors through ''R''.<ref>Milne (2017), Lemma 19.16.</ref> This focuses attention on the representation theory of reductive groups. (To be clear, the representations considered here are representations of ''G'' ''as an algebraic group''. Thus, for a group ''G'' over a field ''k'', the representations are on ''k''-vector spaces, and the action of ''G'' is given by regular functions. It is an important but different problem to classify [[topological group#Representations of compact or locally compact groups|continuous representations]] of the group ''G''('''R''') for a real reductive group ''G'', or similar problems over other fields.)

Chevalley showed that the irreducible representations of a split reductive group over a field ''k'' are finite-dimensional, and they are indexed by [[dominant weight]]s.<ref>Milne (2017), Theorem 22.2.</ref> This is the same as what happens in the representation theory of compact connected Lie groups, or the finite-dimensional representation theory of complex [[semisimple Lie algebra]]s. For ''k'' of characteristic zero, all these theories are essentially equivalent. In particular, every representation of a reductive group ''G'' over a field of characteristic zero is a direct sum of irreducible representations, and if ''G'' is split, the [[character theory|characters]] of the irreducible representations are given by the [[Weyl character formula]]. The [[Borel–Weil theorem]] gives a geometric construction of the irreducible representations of a reductive group ''G'' in characteristic zero, as spaces of sections of [[invertible sheaf|line bundles]] over the flag manifold ''G''/''B''.

The representation theory of reductive groups (other than tori) over a field of positive characteristic ''p'' is less well understood. In this situation, a representation need not be a direct sum of irreducible representations. And although irreducible representations are indexed by dominant weights, the dimensions and characters of the irreducible representations are known only in some cases. {{harvs|txt|last=Andersen, Jantzen and Soergel|year=1994}} determined these characters (proving [[George Lusztig|Lusztig]]'s conjecture) when the characteristic ''p'' is sufficiently large compared to the [[Coxeter number]] of the group. For small primes ''p'', there is not even a precise conjecture.

===Group actions and geometric invariant theory===
An '''[[group-scheme action|action]]''' of a linear algebraic group ''G'' on a variety (or scheme) ''X'' over a field ''k'' is a morphism
:<math>G \times_k X \to X</math>
that satisfies the axioms of a [[Group action (mathematics)|group action]]. As in other types of group theory, it is important to study group actions, since groups arise naturally as symmetries of geometric objects.

Part of the theory of group actions is [[geometric invariant theory]], which aims to construct a quotient variety ''X''/''G'', describing the set of [[orbit (group theory)|orbit]]s of a linear algebraic group ''G'' on ''X'' as an algebraic variety. Various complications arise. For example, if ''X'' is an affine variety, then one can try to construct ''X''/''G'' as [[spectrum of a ring|Spec]] of the [[ring of invariants]] ''O''(''X'')<sup>''G''</sup>. However, [[Masayoshi Nagata]] showed that the ring of invariants need not be finitely generated as a ''k''-algebra (and so Spec of the ring is a scheme but not a variety), a negative answer to [[Hilbert's 14th problem]]. In the positive direction, the ring of invariants is finitely generated if ''G'' is reductive, by [[Haboush's theorem]], proved in characteristic zero by [[David Hilbert|Hilbert]] and Nagata.

Geometric invariant theory involves further subtleties when a reductive group ''G'' acts on a projective variety ''X''. In particular, the theory defines open subsets of "stable" and "semistable" points in ''X'', with the quotient morphism only defined on the set of semistable points.

==Related notions==
Linear algebraic groups admit variants in several directions. Dropping the existence of the inverse map <math>i\colon G \to G</math>, one obtains the notion of a linear algebraic [[monoid]].<ref>{{citation|author=Renner|first=Lex|title=Linear Algebraic Monoids|publisher=Springer|year=2006}}.</ref>

===Lie groups===
For a linear algebraic group ''G'' over the real numbers '''R''', the group of real points ''G''('''R''') is a [[Lie group]], essentially because real polynomials, which describe the multiplication on ''G'', are [[smooth function]]s. Likewise, for a linear algebraic group ''G'' over '''C''', ''G''('''C''') is a [[complex Lie group]]. Much of the theory of algebraic groups was developed by analogy with Lie groups.

There are several reasons why a Lie group may not have the structure of a linear algebraic group over '''R'''.

*A Lie group with an infinite group of components G/G<sup>o</sup> cannot be realized as a linear algebraic group.
*An algebraic group ''G'' over '''R''' may be connected as an algebraic group while the Lie group ''G''('''R''') is not connected, and likewise for [[simply connected]] groups. For example, the algebraic group ''SL''(2) is simply connected over any field, whereas the Lie group ''SL''(2,'''R''') has [[fundamental group]] isomorphic to the integers '''Z'''. The double cover ''H'' of ''SL''(2,'''R'''), known as the '''[[metaplectic group]]''', is a Lie group that cannot be viewed as a linear algebraic group over '''R'''. More strongly, ''H'' has no faithful finite-dimensional representation. 
*[[Anatoly Maltsev]] showed that every simply connected nilpotent Lie group can be viewed as a unipotent algebraic group ''G'' over '''R''' in a unique way.<ref>Milne (2017), Theorem 14.37.</ref> (As a variety, ''G'' is isomorphic to [[affine space]] of some dimension over '''R'''.) By contrast, there are simply connected solvable Lie groups that cannot be viewed as real algebraic groups. For example, the [[universal cover]] ''H'' of the semidirect product ''S''<sup>1</sup> ⋉ '''R'''<sup>2</sup> has center isomorphic to '''Z''', which is not a linear algebraic group, and so ''H'' cannot be viewed as a linear algebraic group over '''R'''.

===Abelian varieties===
[[Algebraic group]]s which are not affine behave very differently. In particular, a smooth connected group scheme which is a projective variety over a field is called an [[abelian variety]]. In contrast to linear algebraic groups, every abelian variety is commutative. Nonetheless, abelian varieties have a rich theory. Even the case of [[elliptic curve]]s (abelian varieties of dimension 1) is central to [[number theory]], with applications including the proof of [[Fermat's Last Theorem]].

===Tannakian categories===
The finite-dimensional representations of an algebraic group ''G'', together with the [[tensor product]] of representations, form a [[tannakian category]] Rep<sub>''G''</sub>. In fact, tannakian categories with a "fiber functor" over a field are equivalent to affine group schemes. (Every affine group scheme over a field ''k'' is ''pro-algebraic'' in the sense that it is an [[inverse limit]] of affine group schemes of finite type over ''k''.<ref>Deligne & Milne (1982), Corollary II.2.7.</ref>) For example, the [[Mumford–Tate group]] and the [[motivic Galois group]] are constructed using this formalism. Certain properties of a (pro-)algebraic group ''G'' can be read from its category of representations. For example, over a field of characteristic zero, Rep<sub>''G''</sub> is a [[semisimple category]] if and only if the identity component of ''G'' is pro-reductive.<ref>Deligne & Milne (1982), Remark II.2.28.</ref>

==See also==
*The [[Group of Lie type|groups of Lie type]] are the finite simple groups constructed from simple algebraic groups over finite fields.
*[[Lang's theorem]]
*[[Generalized flag variety]], [[Bruhat decomposition]], [[BN pair]], [[Weyl group]], [[Cartan subgroup]], [[Adjoint representation#Properties|group of adjoint type]], [[parabolic induction]]
*[[Real form (Lie theory)]], [[Satake diagram]]
*[[Adelic algebraic group]], [[Weil's conjecture on Tamagawa numbers]]
*[[Langlands classification]], [[Langlands program]], [[geometric Langlands program]]
*[[Torsor]], [[nonabelian cohomology]], [[Special group (algebraic group theory)|special group]], [[cohomological invariant]], [[essential dimension]], [[Kneser–Tits conjecture]], [[Serre's conjecture II (algebra)|Serre's conjecture II]]
*[[Pseudo-reductive group]]
*[[Differential Galois theory]]
*[[Distribution on a linear algebraic group]]

==Notes==
{{reflist|30em}}

==References==
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==External links==
* {{springer|title=Linear algebraic group|id=p/l059070}}

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[[Category:Linear algebraic groups| ]]