Editing Reductive group

{{Short description|Concept in mathematics}}
{{Group theory sidebar|Algebraic}}

In [[mathematics]], a '''reductive group''' is a type of [[linear algebraic group]] over a [[field (mathematics)|field]]. One definition is that a connected linear algebraic group ''G'' over a [[perfect field]] is reductive if it has a [[group representation|representation]] that has a finite [[kernel (algebra)|kernel]] and is a [[direct sum]] of [[irreducible representation]]s. Reductive groups include some of the most important groups in mathematics, such as the [[general linear group]] ''GL''(''n'') of [[invertible matrices]], the [[special orthogonal group]] ''SO''(''n''), and the [[symplectic group]] ''Sp''(2''n''). '''Simple algebraic groups''' and (more generally) '''semisimple algebraic groups''' are reductive.

[[Claude Chevalley]] showed that the classification of reductive groups is the same over any [[algebraically closed field]]. In particular, the simple algebraic groups are classified by [[Dynkin diagram]]s, as in the theory of [[compact Lie group]]s or [[complex Lie algebra|complex]] [[semisimple Lie algebra]]s. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the [[real number]]s '''R''' or a [[number field]], the classification is well understood. The [[classification of finite simple groups]] says that most finite simple groups arise as the group ''G''(''k'') of ''k''-[[rational point]]s of a simple algebraic group ''G'' over a [[finite field]] ''k'', or as minor variants of that construction.

Reductive groups have a rich [[representation theory]] in various contexts. First, one can study the representations of a reductive group ''G'' over a field ''k'' as an algebraic group, which are actions of ''G'' on ''k''-vector spaces. But also, one can study the complex representations of the group ''G''(''k'') when ''k'' is a finite field, or the infinite-dimensional [[unitary representation]]s of a real reductive group, or the [[automorphic representation]]s of an [[adelic algebraic group]]. The structure theory of reductive groups is used in all these areas.

==Definitions==
{{main|Linear algebraic group}}
A [[linear algebraic group]] over a field ''k'' is defined as a [[smooth scheme|smooth]] closed [[group scheme|subgroup scheme]] of ''GL''(''n'') over ''k'', for some positive integer ''n''. Equivalently, a linear algebraic group over ''k'' is a smooth [[affine scheme|affine]] group scheme over ''k''.

=== With the unipotent radical ===
A [[connected space|connected]] linear algebraic group <math>G</math> over an algebraically closed field is called '''semisimple''' if every smooth connected [[solvable group|solvable]] [[normal subgroup]] of <math>G</math> is trivial. More generally, a connected linear algebraic group <math>G</math> over an algebraically closed field is called '''reductive''' if the largest smooth connected [[linear algebraic group#Unipotent groups|unipotent]] normal subgroup of <math>G</math> is trivial.<ref>SGA 3 (2011), v. 3, Définition XIX.1.6.1.</ref> This normal subgroup is called the '''unipotent radical''' and is denoted <math>R_u(G)</math>. (Some authors do not require reductive groups to be connected.) A group <math>G</math> over an arbitrary field ''k'' is called semisimple or reductive if the [[fiber product of schemes|base change]] <math>G_{\overline k}</math> is semisimple or reductive, where <math>\overline k</math> is an [[algebraic closure]] of ''k''. (This is equivalent to the definition of reductive groups in the introduction when ''k'' is perfect.<ref>Milne (2017), Proposition 21.60.</ref>) Any [[linear algebraic group#Tori|torus]] over ''k'', such as the [[linear algebraic group#Examples|multiplicative group]] ''G''<sub>''m''</sub>, is reductive.

=== With representation theory ===
Over fields of characteristic zero another equivalent definition of a reductive group is a connected group <math>G</math> admitting a faithful semisimple representation which remains semisimple over its algebraic closure <math>k^{al}</math><ref>{{Cite book|last=Milne|url=https://www.jmilne.org/math/CourseNotes/iAG200.pdf|title=Linear Algebraic Groups|pages=381–394}}</ref> <sup>page 424</sup>.

=== Simple reductive groups ===
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 (group theory)|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 [[multiplicative group#Group scheme of roots of unity|group scheme μ<sub>''n''</sub>]] of ''n''th roots of unity.

A '''central isogeny''' of reductive groups is a surjective [[group homomorphism|homomorphism]] with kernel a finite [[central subgroup]] scheme. Every reductive group over a field admits a central isogeny from the product of a torus and some simple groups. For example, over any field ''k'',
:<math>GL(n)\cong (G_m\times SL(n))/\mu_n.</math>

It is slightly awkward that the definition of a reductive group over a field involves passage to the algebraic closure. For a perfect field ''k'', that can be avoided: a linear algebraic group ''G'' over ''k'' is reductive if and only if every smooth connected unipotent normal ''k''-subgroup of ''G'' is trivial. For an arbitrary field, the latter property defines a [[pseudo-reductive group]], which is somewhat more general.

=== Split-reductive groups ===
A reductive group ''G'' over a field ''k'' is called '''split''' if it contains a split maximal torus ''T'' over ''k'' (that is, a [[linear algebraic group#Tori|split torus]] in ''G'' whose base change to <math>\overline k</math> is a [[maximal torus]] in <math>G_{\overline k}</math>). It is equivalent to say that ''T'' is a split torus in ''G'' that is maximal among all ''k''-tori in ''G''.<ref>Borel (1991), 18.2(i).</ref> These kinds of groups are useful because their classification can be described through combinatorical data called root data.

==Examples==

=== GL<sub>''n''</sub> and SL<sub>''n''</sub> ===
A fundamental example of a reductive group is the '''general linear group''' <math>\text{GL}_n</math> of invertible ''n'' × ''n'' matrices over a field ''k'', for a natural number ''n''. In particular, the '''multiplicative group''' ''G''<sub>''m''</sub> is the group ''GL''(1), and so its group ''G''<sub>''m''</sub>(''k'') of ''k''-rational points is the group ''k''* of nonzero elements of ''k'' under multiplication. Another reductive group is the [[special linear group]] ''SL''(''n'') over a field ''k'', the subgroup of matrices with [[determinant]] 1. In fact, ''SL''(''n'') is a simple algebraic group for ''n'' at least 2.

=== O(''n''), SO(''n''), and Sp(''n'') ===
An important simple group is the [[symplectic group]] ''Sp''(2''n'') over a field ''k'', the subgroup of ''GL''(2''n'') that preserves a nondegenerate alternating [[bilinear form]] on the [[vector space]] ''k''<sup>2''n''</sup>. Likewise, the [[orthogonal group]] ''O''(''q'') is the subgroup of the general linear group that preserves a nondegenerate [[quadratic form]] ''q'' on a vector space over a field ''k''. The algebraic group ''O''(''q'') has two [[connected component (topology)|connected components]], and its [[identity component]] ''SO''(''q'') is reductive, in fact simple for ''q'' of dimension ''n'' at least 3. (For ''k'' of characteristic 2 and ''n'' odd, the group scheme ''O''(''q'') is in fact connected but not smooth over ''k''. The simple group ''SO''(''q'') can always be defined as the maximal smooth connected subgroup of ''O''(''q'') over ''k''.) When ''k'' is algebraically closed, any two (nondegenerate) quadratic forms of the same dimension are isomorphic, and so it is reasonable to call this group ''SO''(''n''). For a general field ''k'', different quadratic forms of dimension ''n'' can yield non-isomorphic simple groups ''SO''(''q'') over ''k'', although they all have the same base change to the algebraic closure <math>\overline k</math>.

=== Tori ===
The group <math>\mathbb{G}_m</math> and products of it are called the [[Algebraic torus|algebraic tori]]. They are examples of reductive groups since they embed in <math>\text{GL}_n</math> through the diagonal, and from this representation, their unipotent radical is trivial. For example, <math>\mathbb{G}_m\times \mathbb {G}_m</math> embeds in <math>\text{GL}_2</math> from the map<blockquote><math>(a_1,a_2) \mapsto \begin{bmatrix}
a_1 & 0 \\
0 & a_2
\end{bmatrix}.</math></blockquote>

=== Non-examples ===

* Any [[unipotent group]] is not reductive since its unipotent radical is itself. This includes the additive group <math>\mathbb{G}_a</math>.
* The [[Borel group]] <math>B_n</math> of <math>\text{GL}_n</math> has a non-trivial unipotent radical <math>\mathbb{U}_n</math> of upper-triangular matrices with <math>1</math> on the diagonal. This is an example of a non-reductive group which is not unipotent.

==== Associated reductive group ====
Note that the normality of the unipotent radical <math>R_u(G)</math> implies that the quotient group <math>G/R_u(G)</math> is reductive. For example,<blockquote><math>B_n/(R_u(B_n)) \cong \prod^n_{i=1} \mathbb{G}_m.</math></blockquote>

==Other characterizations of reductive groups==
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. For a compact Lie group ''K'' with complexification ''G'', the inclusion from ''K'' into the complex reductive group ''G''('''C''') is a [[homotopy equivalence]], with respect to the classical topology on ''G''('''C'''). For example, the inclusion from the [[unitary group]] ''U''(''n'') to ''GL''(''n'','''C''') is a homotopy equivalence.

For a reductive group ''G'' over a field of [[characteristic of a field|characteristic]] zero, all finite-dimensional representations of ''G'' (as an algebraic group) are [[semisimple representation|completely reducible]], that is, they are direct sums of irreducible representations.<ref>Milne (2017), Theorem 22.42.</ref> That is the source of the name "reductive". Note, however, that complete reducibility fails for reductive groups in positive characteristic (apart from tori). In more detail: an affine group scheme ''G'' of [[Glossary of scheme theory#finite type (locally)|finite type]] over a field ''k'' is called '''linearly reductive''' if its finite-dimensional representations are completely reducible. For ''k'' of characteristic zero, ''G'' is linearly reductive if and only if the identity component ''G''<sup>o</sup> of ''G'' is reductive.<ref>Milne (2017), Corollary 22.43.</ref> For ''k'' of characteristic ''p''>0, however, [[Masayoshi Nagata]] showed that ''G'' is linearly reductive if and only if ''G''<sup>o</sup> is of [[Group scheme#Constructions|multiplicative type]] and ''G''/''G''<sup>o</sup> has order prime to ''p''.<ref>Demazure & Gabriel (1970), Théorème IV.3.3.6.</ref>

==Roots==
The classification of reductive algebraic groups is in terms of the associated [[root system]], as in the theories of complex semisimple Lie algebras or compact Lie groups. Here is the way roots appear for reductive groups.

Let ''G'' be a split reductive group over a field ''k'', and let ''T'' be a split maximal torus in ''G''; so ''T'' is isomorphic to (''G''<sub>''m''</sub>)<sup>''n''</sup> for some ''n'', with ''n'' called the '''rank''' of ''G''. Every representation of ''T'' (as an algebraic group) is a direct sum of 1-dimensional representations.<ref>Milne (2017), Theorem 12.12.</ref> A '''weight''' for ''G'' means an isomorphism class of 1-dimensional representations of ''T'', or equivalently a homomorphism ''T'' → ''G''<sub>''m''</sub>. The weights form a group ''X''(''T'') under [[tensor product]] of representations, with ''X''(''T'') isomorphic to the product of ''n'' copies of the [[integer]]s, '''Z'''<sup>''n''</sup>.

The [[adjoint representation]] is the action of ''G'' by conjugation on its [[linear algebraic group#The Lie algebra of an algebraic group|Lie algebra]] <math>\mathfrak g</math>. A '''root''' of ''G'' means a nonzero weight that occurs in the action of ''T'' ⊂ ''G'' on <math>\mathfrak g</math>. The subspace of <math>\mathfrak g</math> corresponding to each root is 1-dimensional, and the subspace of <math>\mathfrak g</math> fixed by ''T'' is exactly the Lie algebra <math>\mathfrak t</math> of ''T''.<ref name="M2111">Milne (2017), Theorem 21.11.</ref> Therefore, the Lie algebra of ''G'' decomposes into <math>\mathfrak t</math> together with 1-dimensional subspaces indexed by the set Φ of roots:
:<math>{\mathfrak g} = {\mathfrak t}\oplus \bigoplus_{\alpha\in\Phi} {\mathfrak g}_{\alpha}.</math>

For example, when ''G'' is the group ''GL''(''n''), its Lie algebra <math>{\mathfrak gl}(n)</math> is the vector space of all ''n'' × ''n'' matrices over ''k''. Let ''T'' be the subgroup of diagonal matrices in ''G''. Then the root-space decomposition expresses <math>{\mathfrak gl}(n)</math> as the direct sum of the diagonal matrices and the 1-dimensional subspaces indexed by the off-diagonal positions (''i'', ''j''). Writing ''L''<sub>1</sub>,...,''L''<sub>''n''</sub> for the standard basis for the weight lattice ''X''(''T'') ≅ '''Z'''<sup>''n''</sup>, the roots are the elements ''L''<sub>''i''</sub> − ''L''<sub>''j''</sub> for all ''i'' ≠ ''j'' from 1 to ''n''.

The roots of a semisimple group form a '''root system'''; this is a combinatorial structure which can be completely classified. More generally, the roots of a reductive group form a [[root datum]], a slight variation.<ref>Milne (2017), Corollary 21.12.</ref> The '''[[Weyl group]]''' of a reductive group ''G'' means the [[quotient group]] of the [[normalizer]] of a maximal torus by the torus, ''W'' = ''N''<sub>''G''</sub>(''T'')/''T''. The Weyl group is in fact a finite group generated by reflections. For example, for the group ''GL''(''n'') (or ''SL''(''n'')), the Weyl group is the [[symmetric group]] ''S''<sub>''n''</sub>.

There are finitely many [[Borel subgroup]]s containing a given maximal torus, and they are permuted [[simply transitively]] by the Weyl group (acting by [[Conjugacy class#Conjugacy of subgroups and general subsets|conjugation]]).<ref>Milne (2017), Proposition 17.53.</ref> A choice of Borel subgroup determines a set of '''positive roots''' Φ<sup>+</sup> ⊂ Φ, with the property that Φ is the disjoint union of Φ<sup>+</sup> and −Φ<sup>+</sup>. Explicitly, the Lie algebra of ''B'' is the direct sum of the Lie algebra of ''T'' and the positive root spaces:
:<math>{\mathfrak b}={\mathfrak t}\oplus \bigoplus_{\alpha\in\Phi^{+}} {\mathfrak g}_{\alpha}.</math>
For example, if ''B'' is the Borel subgroup of upper-triangular matrices in ''GL''(''n''), then this is the obvious decomposition of the subspace <math>\mathfrak b</math> of upper-triangular matrices in <math>{\mathfrak gl}(n)</math>. The positive roots are ''L''<sub>''i''</sub> − ''L''<sub>''j''</sub> for 1 ≤ ''i'' < ''j'' ≤ ''n''.

A '''simple root''' means a positive root that is not a sum of two other positive roots. Write Δ for the set of simple roots. The number ''r'' of simple roots is equal to the rank of the [[commutator subgroup]] of ''G'', called the '''semisimple rank''' of ''G'' (which is simply the rank of ''G'' if ''G'' is semisimple). For example, the simple roots for ''GL''(''n'') (or ''SL''(''n'')) are ''L''<sub>''i''</sub> − ''L''<sub>''i''+1</sub> for 1 ≤ ''i'' ≤ ''n'' − 1.

Root systems are classified by the corresponding [[Dynkin diagram]], which is a finite [[graph (discrete mathematics)|graph]] (with some edges directed or multiple). The set of vertices of the Dynkin diagram is the set of simple roots. In short, the Dynkin diagram describes the angles between the simple roots and their relative lengths, with respect to a Weyl group-invariant [[inner product]] on the weight lattice. The connected Dynkin diagrams (corresponding to simple groups) are pictured below.

For a split reductive group ''G'' over a field ''k'', an important point is that a root α determines not just a 1-dimensional subspace of the Lie algebra of ''G'', but also a copy of the additive group ''G''<sub>a</sub> in ''G'' with the given Lie algebra, called a '''root subgroup''' ''U''<sub>α</sub>. The root subgroup is the unique copy of the additive group in ''G'' which is [[normalizer|normalized]] by ''T'' and which has the given Lie algebra.<ref name = "M2111" /> The whole group ''G'' is generated (as an algebraic group) by ''T'' and the root subgroups, while the Borel subgroup ''B'' is generated by ''T'' and the positive root subgroups. In fact, a split semisimple group ''G'' is generated by the root subgroups alone.

==Parabolic subgroups==
For a split reductive group ''G'' over a field ''k'', the smooth connected subgroups of ''G'' that contain a given Borel subgroup ''B'' of ''G'' are in one-to-one correspondence with the subsets of the set Δ of simple roots (or equivalently, the subsets of the set of vertices of the Dynkin diagram). Let ''r'' be the order of Δ, the semisimple rank of ''G''. Every '''parabolic subgroup''' of ''G'' is [[conjugacy class#Conjugacy of subgroups and general subsets|conjugate]] to a subgroup containing ''B'' by some element of ''G''(''k''). As a result, there are exactly 2<sup>''r''</sup> conjugacy classes of parabolic subgroups in ''G'' over ''k''.<ref>Borel (1991), Proposition 21.12.</ref> Explicitly, the parabolic subgroup corresponding to a given subset ''S'' of Δ is the group generated by ''B'' together with the root subgroups ''U''<sub>−α</sub> for α in ''S''. For example, the parabolic subgroups of ''GL''(''n'') that contain the Borel subgroup ''B'' above are the groups of invertible matrices with zero entries below a given set of squares along the diagonal, such as:
:<math>\left \{ \begin{bmatrix}
 * & * & * & *\\
 * & * & * & *\\
 0 & 0 & * & *\\
 0 & 0 & 0 & *
\end{bmatrix} \right \}</math>

By definition, a '''parabolic subgroup''' ''P'' of a reductive group ''G'' over a field ''k'' is a smooth ''k''-subgroup such that the quotient variety ''G''/''P'' is [[proper scheme|proper]] over ''k'', or equivalently [[projective variety|projective]] over ''k''. Thus the classification of parabolic subgroups amounts to a classification of the [[generalized flag variety|projective homogeneous varieties]] for ''G'' (with smooth stabilizer group; that is no restriction for ''k'' of characteristic zero). For ''GL''(''n''), these are the '''flag varieties''', parametrizing sequences of linear subspaces of given dimensions ''a''<sub>1</sub>,...,''a''<sub>''i''</sub> contained in a fixed vector space ''V'' of dimension ''n'':
:<math>0\subset S_{a_1}\subset \cdots \subset S_{a_i}\subset V.</math>
For the orthogonal group or the symplectic group, the projective homogeneous varieties have a similar description as varieties of [[isotropic subspace|isotropic]] flags with respect to a given quadratic form or symplectic form. For any reductive group ''G'' with a Borel subgroup ''B'', ''G''/''B'' is called the '''flag variety''' or '''flag manifold''' of ''G''.

==Classification of split reductive groups==
[[File:Finite Dynkin diagrams.svg|480px|thumb|The connected Dynkin diagrams]]

Chevalley showed in 1958 that the reductive groups over any algebraically closed field are classified up to isomorphism by root data.<ref>Chevalley (2005); Springer (1998), 9.6.2 and 10.1.1.</ref> In particular, the semisimple groups over an algebraically closed field are classified up to central isogenies by their Dynkin diagram, and the simple groups correspond to the connected diagrams. Thus there are simple groups of types A<sub>''n''</sub>,  B<sub>''n''</sub>, C<sub>''n''</sub>,  D<sub>''n''</sub>,  E<sub>6</sub>, E<sub>7</sub>, E<sub>8</sub>, F<sub>4</sub>, G<sub>2</sub>. This result is essentially identical to the classifications of compact Lie groups or complex semisimple Lie algebras, by [[Wilhelm Killing]] and [[Élie Cartan]] in the 1880s and 1890s. In particular, the dimensions, centers, and other properties of the simple algebraic groups can be read from the [[list of simple Lie groups]]. It is remarkable that the classification of reductive groups is independent of the characteristic. For comparison, there are many more simple Lie algebras in positive characteristic than in characteristic zero.

The [[exceptional group]]s ''G'' of type G<sub>2</sub> and E<sub>6</sub> had been constructed earlier, at least in the form of the abstract group ''G''(''k''), by [[Leonard Eugene Dickson|L. E. Dickson]]. For example, the group ''G''<sub>2</sub> is the [[automorphism group]] of an [[octonion algebra]] over ''k''. By contrast, the Chevalley groups of type F<sub>4</sub>, E<sub>7</sub>, E<sub>8</sub> over a field of positive characteristic were completely new.

More generally, the classification of ''split'' reductive groups is the same over any field.<ref>Milne (2017), Theorems 23.25 and 23.55.</ref> A semisimple group ''G'' over a field ''k'' is called '''simply connected''' if every central isogeny from a semisimple group to ''G'' is an isomorphism. (For ''G'' semisimple over the [[complex number]]s, being simply connected in this sense is equivalent to ''G''('''C''') being [[simply connected]] in the classical topology.) Chevalley's classification gives that, over any field ''k'', there is a unique simply connected split semisimple group ''G'' with a given Dynkin diagram, with simple groups corresponding to the connected diagrams. At the other extreme, a semisimple group is of '''adjoint type''' if its center is trivial. The split semisimple groups over ''k'' with given Dynkin diagram are exactly the groups ''G''/''A'', where ''G'' is the simply connected group and ''A'' is a ''k''-subgroup scheme of the center of ''G''.

For example, the simply connected split simple groups over a field ''k'' corresponding to the "classical" Dynkin diagrams are as follows:
*A<sub>''n''</sub>: ''SL''(''n''+1) over ''k'';
*B<sub>''n''</sub>: the [[spin group]] Spin(2''n''+1) associated to a quadratic form of dimension 2''n''+1 over ''k'' with [[Witt index]] ''n'', for example the form
::<math>q(x_1,\ldots,x_{2n+1})=x_1x_2+x_3x_4+\cdots+x_{2n-1}x_{2n}+x_{2n+1}^2;</math>
*C<sub>''n''</sub>: the symplectic group ''Sp''(2''n'') over ''k'';
*D<sub>''n''</sub>: the spin group Spin(2''n'') associated to a quadratic form of dimension 2''n'' over ''k'' with Witt index ''n'', which can be written as:
::<math>q(x_1,\ldots,x_{2n})=x_1x_2+x_3x_4+\cdots+x_{2n-1}x_{2n}.</math>

The [[outer automorphism group]] of a split reductive group ''G'' over a field ''k'' is isomorphic to the automorphism group of the root datum of ''G''. Moreover, the automorphism group of ''G'' splits as a [[semidirect product]]:
:<math>\operatorname{Aut}(G)\cong \operatorname{Out}(G)\ltimes (G/Z)(k),</math>
where ''Z'' is the center of ''G''.<ref>Milne (2017), Corollary 23.47.</ref> For a split semisimple simply connected group ''G'' over a field, the outer automorphism group of ''G'' has a simpler description: it is the automorphism group of the Dynkin diagram of ''G''.

==Reductive group schemes==
A [[group scheme]] ''G'' over a scheme ''S'' is called '''reductive''' if the morphism ''G'' → ''S'' is [[smooth morphism|smooth]] and affine, and every geometric fiber <math>G_{\overline k}</math> is reductive. (For a point ''p'' in ''S'', the corresponding geometric fiber means the base change of ''G'' to an algebraic closure <math>\overline k</math> of the residue field of ''p''.) Extending Chevalley's work, [[Michel Demazure]] and Grothendieck showed that '''pinned''' reductive group schemes over any nonempty scheme ''S'' are classified by root data.<ref>SGA 3 (2011), v. 3, Théorème XXV.1.1; Conrad (2014), Theorems 6.1.16 and 6.1.17.</ref> This statement includes the existence of Chevalley groups as group schemes over '''Z''', and it says that every pinned reductive group over a scheme ''S'' is isomorphic to the base change of a Chevalley group from '''Z''' to ''S''. A pinning of a split reductive group is a choice of root basis and also a choice of trivialisation of the one-dimensional additive group corresponding to each simple root. This statement is false without the pinning; for example, suppose that ''A'' is a Dedekind domain and that ''I'' is an ideal in ''A'' whose class in the class group of ''A'' is not a square. Then ''SL(A + I)'' and SL_2(A) are split and reductive over ''Spec A'' and have the same root data but they are not isomorphic: the flag scheme (the quotient by a Borel subgroup scheme) of the first is the projective line bundle '''P'''''(A + I)'' and has no section with trivial normal bundle (a section corresponds to a short exact sequence ''0 → J → A + I → K → 0'' where ''J, K'' are ideal classes and the normal bundle is then ''J^{-1}K'', which is not trivial since ''JK'' is isomorphic to ''I'') while the flag scheme of the second is '''P'''''^1_A'' and does possess sections with trivial normal bundle.

==Real reductive groups==
In the context of [[Lie group]]s rather than algebraic groups, a '''real reductive group''' is a Lie group ''G'' such that there is a linear algebraic group ''L'' over '''R''' whose identity component (in the [[Zariski topology]]) is reductive, and a homomorphism ''G'' → ''L''('''R''') whose kernel is finite and whose image is open in ''L''('''R''') (in the classical topology). It is also standard to assume that the image of the adjoint representation Ad(''G'') is contained in Int(''g''<sub>'''C'''</sub>) = Ad(''L''<sup>0</sup>('''C''')) (which is automatic for ''G'' connected).<ref>Springer (1979), section 5.1.</ref>

In particular, every connected semisimple Lie group (meaning that its Lie algebra is semisimple) is reductive. Also, the Lie group '''R''' is reductive in this sense, since it can be viewed as the identity component of ''GL''(1,'''R''') &cong; '''R'''*. The problem of classifying the real reductive groups largely reduces to classifying the simple Lie groups. These are classified by their [[Satake diagram]]; or one can just refer to the [[list of simple Lie groups]] (up to finite coverings).

Useful theories of [[admissible representation]]s and unitary representations have been developed for real reductive groups in this generality. The main differences between this definition and the definition of a reductive algebraic group have to do with the fact that 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 [[projective linear group]] ''PGL''(2) is connected as an algebraic group over any field, but its group of real points ''PGL''(2,'''R''') has two connected components. The identity component of ''PGL''(2,'''R''') (sometimes called ''PSL''(2,'''R''')) is a real reductive group that cannot be viewed as an algebraic group. Similarly, ''SL''(2) is simply connected as an algebraic group over any field, but the Lie group ''SL''(2,'''R''') has [[fundamental group]] isomorphic to the integers '''Z''', and so ''SL''(2,'''R''') has nontrivial [[covering space]]s. By definition, all finite coverings of ''SL''(2,'''R''') (such as the [[metaplectic group]]) are real reductive groups. On the other hand, the [[universal cover]] of ''SL''(2,'''R''') is not a real reductive group, even though its Lie algebra is [[reductive Lie algebra|reductive]], that is, the product of a semisimple Lie algebra and an abelian Lie algebra.

For a connected real reductive group ''G'', the quotient manifold ''G''/''K'' of ''G'' by a [[maximal compact subgroup]] ''K'' is a [[symmetric space]] of non-compact type. In fact, every symmetric space of non-compact type arises this way. These are central examples in [[Riemannian geometry]] of manifolds with nonpositive [[sectional curvature]]. For example, ''SL''(2,'''R''')/''SO''(2) is the [[hyperbolic plane]], and ''SL''(2,'''C''')/''SU''(2) is hyperbolic 3-space.

For a reductive group ''G'' over a field ''k'' that is complete with respect to a [[discrete valuation]] (such as the [[p-adic numbers]] '''Q'''<sub>''p''</sub>), the '''[[building (mathematics)|affine building]]''' ''X'' of ''G'' plays the role of the symmetric space. Namely, ''X'' is a [[simplicial complex]] with an action of ''G''(''k''), and ''G''(''k'') preserves a [[CAT(0)]] metric on ''X'', the analog of a metric with nonpositive curvature. The dimension of the affine building is the ''k''-rank of ''G''. For example, the building of ''SL''(2,'''Q'''<sub>''p''</sub>) is a [[tree (graph theory)|tree]].

==Representations of reductive groups==
For a split reductive group ''G'' over a field ''k'', the irreducible representations of ''G'' (as an algebraic group) are parametrized by the [[dominant weight]]s, which are defined as the intersection of the weight lattice ''X''(''T'')  ≅ '''Z'''<sup>''n''</sup> with a convex cone (a [[Weyl chamber]]) in '''R'''<sup>''n''</sup>. In particular, this parametrization is independent of the characteristic of ''k''. In more detail, fix a split maximal torus and a Borel subgroup, ''T'' ⊂ ''B'' ⊂ ''G''. Then ''B'' is the semidirect product of ''T'' with a smooth connected unipotent subgroup ''U''. Define a '''highest weight vector''' in a representation ''V'' of ''G'' over ''k'' to be a nonzero vector ''v'' such that ''B'' maps the line spanned by ''v'' into itself. Then ''B'' acts on that line through its quotient group ''T'', by some element λ of the weight lattice ''X''(''T''). Chevalley showed that every irreducible representation of ''G'' has a unique highest weight vector up to scalars; the corresponding "highest weight" λ is dominant; and every dominant weight λ is the highest weight of a unique irreducible representation ''L''(λ) of ''G'', up to isomorphism.<ref>Milne (2017), Theorem 22.2.</ref>

There remains the problem of describing the irreducible representation with given highest weight. For ''k'' of characteristic zero, there are essentially complete answers. For a dominant weight λ, define the '''Schur module''' ∇(λ) as the ''k''-vector space of sections of the ''G''-equivariant [[invertible sheaf|line bundle]] on the flag manifold ''G''/''B'' associated to λ; this is a representation of ''G''. For ''k'' of characteristic zero, the [[Borel–Weil theorem]] says that the irreducible representation ''L''(λ) is isomorphic to the Schur module ∇(λ). Furthermore, the [[Weyl character formula]] gives the [[character theory|character]] (and in particular the dimension) of this representation.

For a split reductive group ''G'' over a field ''k'' of positive characteristic, the situation is far more subtle, because representations of ''G'' are typically not direct sums of irreducibles. For a dominant weight λ, the irreducible representation ''L''(λ) is the unique simple submodule (the [[socle (mathematics)|socle]]) of the Schur module ∇(λ), but it need not be equal to the Schur module. The dimension and character of the Schur module are given by the Weyl character formula (as in characteristic zero), by [[George Kempf]].<ref>Jantzen (2003), Proposition II.4.5 and Corollary II.5.11.</ref> The dimensions and characters of the irreducible representations ''L''(λ) are in general unknown, although a large body of theory has been developed to analyze these representations. One important result is that the dimension and character of ''L''(λ) are known when the characteristic ''p'' of ''k'' is much bigger than the [[Coxeter number]] of ''G'', by [[Henning Haahr Andersen|Henning Andersen]], [[Jens Carsten Jantzen|Jens Jantzen]], and Wolfgang Soergel (proving [[George Lusztig|Lusztig]]'s conjecture in that case). Their character formula for ''p'' large is based on the [[Kazhdan–Lusztig polynomial]]s, which are combinatorially complex.<ref>Jantzen (2003), section II.8.22.</ref> For any prime ''p'', Simon Riche and [[Geordie Williamson]] conjectured the irreducible characters of a reductive group in terms of the ''p''-Kazhdan-Lusztig polynomials, which are even more complex, but at least are computable.<ref>Riche & Williamson (2018), section 1.8.</ref>

==Non-split reductive groups==
As discussed above, 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. Some examples among the [[classical group]]s are:
*Every nondegenerate quadratic form ''q'' over a field ''k'' determines a reductive group G = ''SO''(''q''). Here ''G'' is simple if ''q'' has dimension ''n'' at least 3, since <math>G_{\overline k}</math> is isomorphic to ''SO''(''n'') over an algebraic closure <math>\overline k</math>. The ''k''-rank of ''G'' is equal to the '''Witt index''' of ''q'' (the maximum dimension of an isotropic subspace over ''k'').<ref name="B234">Borel (1991), section 23.4.</ref> So the simple group ''G'' is split over ''k'' if and only if ''q'' has the maximum possible Witt index, <math>\lfloor n/2\rfloor</math>.
*Every [[central simple algebra]] ''A'' over ''k'' determines a reductive group ''G'' = ''SL''(1,''A''), the kernel of the [[reduced norm]] on the [[group of units]] ''A''* (as an algebraic group over ''k''). The '''degree''' of ''A'' means the square root of the dimension of ''A'' as a ''k''-vector space. Here ''G'' is simple if ''A'' has degree ''n'' at least 2, since <math>G_{\overline k}</math> is isomorphic to ''SL''(''n'') over <math>\overline k</math>. If ''A'' has index ''r'' (meaning that ''A'' is isomorphic to the matrix algebra ''M''<sub>''n''/''r''</sub>(''D'') for a [[division algebra]] ''D'' of degree ''r'' over ''k''), then the ''k''-rank of ''G'' is (''n''/''r'') − 1.<ref>Borel (1991), section 23.2.</ref> So the simple group ''G'' is split over ''k'' if and only if ''A'' is a matrix algebra over ''k''.

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 fields, but for arbitrary fields there are many open questions.

A reductive group over a field ''k'' is called '''isotropic''' if it has ''k''-rank greater than 0 (that is, if it contains a nontrivial split torus), and otherwise '''anisotropic'''. For a semisimple group ''G'' over a field ''k'', the following conditions are equivalent:
*''G'' is isotropic (that is, ''G'' contains a copy of the multiplicative group ''G''<sub>''m''</sub> over ''k'');
*''G'' contains a parabolic subgroup over ''k'' not equal to ''G'';
*''G'' contains a copy of the additive group ''G''<sub>''a''</sub> over ''k''.
For ''k'' perfect, it is also equivalent to say that ''G''(''k'') contains a [[Linear algebraic group#Semisimple and unipotent elements|unipotent]] element other than 1.<ref>Borel & Tits (1971), Corollaire 3.8.</ref>

For a connected linear algebraic group ''G'' over a local field ''k'' of characteristic zero (such as the real numbers), the group ''G''(''k'') is [[compact space|compact]] in the classical topology (based on the topology of ''k'') if and only if ''G'' is reductive and anisotropic.<ref>Platonov & Rapinchuk (1994), Theorem 3.1.</ref> Example: the orthogonal group [[indefinite orthogonal group|''SO''(''p'',''q'')]] over '''R''' has real rank min(''p'',''q''), and so it is anisotropic if and only if ''p'' or ''q'' is zero.<ref name = "B234" />

A reductive group ''G'' over a field ''k'' is called '''quasi-split''' if it contains a Borel subgroup over ''k''. A split reductive group is quasi-split. If ''G'' is quasi-split over ''k'', then any two Borel subgroups of ''G'' are conjugate by some element of ''G''(''k'').<ref>Borel (1991), Theorem 20.9(i).</ref> Example: the orthogonal group ''SO''(''p'',''q'') over '''R''' is split if and only if |''p''−''q''| ≤ 1, and it is quasi-split if and only if |''p''−''q''| ≤ 2.<ref name = "B234" />

==Structure of semisimple groups as abstract groups==
For a simply connected split semisimple group ''G'' over a field ''k'', [[Robert Steinberg]] gave an explicit [[presentation of a group|presentation]] of the abstract group ''G''(''k'').<ref>Steinberg (2016), Theorem 8.</ref> It is generated by copies of the additive group of ''k'' indexed by the roots of ''G'' (the root subgroups), with relations determined by the Dynkin diagram of ''G''.

For a simply connected split semisimple group ''G'' over a perfect field ''k'', Steinberg also determined the automorphism group of the abstract group ''G''(''k''). Every automorphism is the product of an [[inner automorphism]], a diagonal automorphism (meaning conjugation by a suitable <math>\overline k</math>-point of a maximal torus), a graph automorphism (corresponding to an automorphism of the Dynkin diagram), and a field automorphism (coming from an automorphism of the field ''k'').<ref>Steinberg (2016), Theorem 30.</ref>

For a ''k''-simple algebraic group ''G'', '''Tits's simplicity theorem''' says that the abstract group ''G''(''k'') is close to being simple, under mild assumptions. Namely, suppose that ''G'' is isotropic over ''k'', and suppose that the field ''k'' has at least 4 elements. Let ''G''(''k'')<sup>+</sup> be the subgroup of the abstract group ''G''(''k'') generated by ''k''-points of copies of the additive group ''G''<sub>''a''</sub> over ''k'' contained in ''G''. (By the assumption that ''G'' is isotropic over ''k'', the group ''G''(''k'')<sup>+</sup> is nontrivial, and even Zariski dense in ''G'' if ''k'' is infinite.) Then the quotient group of ''G''(''k'')<sup>+</sup> by its center is simple (as an abstract group).<ref>Tits (1964), Main Theorem; Gille (2009), Introduction.</ref> The proof uses [[Jacques Tits]]'s machinery of [[BN-pair]]s.

The exceptions for fields of order 2 or 3 are well understood. For ''k'' = '''F'''<sub>2</sub>, Tits's simplicity theorem remains valid except when ''G'' is split of type ''A''<sub>1</sub>, ''B''<sub>2</sub>, or ''G''<sub>2</sub>, or non-split (that is, unitary) of type ''A''<sub>2</sub>. For ''k'' = '''F'''<sub>3</sub>, the theorem holds except for ''G'' of type ''A''<sub>1</sub>.<ref>Tits (1964), section 1.2.</ref>

For a ''k''-simple group ''G'', in order to understand the whole group ''G''(''k''), one can consider the '''Whitehead group''' ''W''(''k'',''G'')=''G''(''k'')/''G''(''k'')<sup>+</sup>. For ''G'' simply connected and quasi-split, the Whitehead group is trivial, and so the whole group ''G''(''k'') is simple modulo its center.<ref>Gille (2009), Théorème 6.1.</ref> More generally, the [[Kneser–Tits problem]] asks for which isotropic ''k''-simple groups the Whitehead group is trivial. In all known examples, ''W''(''k'',''G'') is abelian.

For an anisotropic ''k''-simple group ''G'', the abstract group ''G''(''k'') can be far from simple. For example, let ''D'' be a division algebra with center a ''p''-adic field ''k''. Suppose that the dimension of ''D'' over ''k'' is finite and greater than 1. Then ''G'' = ''SL''(1,''D'') is an anisotropic ''k''-simple group. As mentioned above, ''G''(''k'') is compact in the classical topology. Since it is also [[totally disconnected]], ''G''(''k'') is a [[profinite group]] (but not finite). As a result, ''G''(''k'') contains infinitely many normal subgroups of finite [[index of a subgroup|index]].<ref>Platonov & Rapinchuk (1994), section 9.1.</ref>

==Lattices and arithmetic groups==
Let ''G'' be a linear algebraic group over the [[rational number]]s '''Q'''. Then ''G'' can be extended to an affine group scheme ''G'' over '''Z''', and this determines an abstract group ''G''('''Z'''). An '''[[arithmetic group]]''' means any subgroup of ''G''('''Q''') that is [[commensurability (group theory)|commensurable]] with ''G''('''Z'''). (Arithmeticity of a subgroup of ''G''('''Q''') is independent of the choice of '''Z'''-structure.) For example, ''SL''(''n'','''Z''') is an arithmetic subgroup of ''SL''(''n'','''Q''').

For a Lie group ''G'', a '''[[lattice (discrete subgroup)|lattice]]''' in ''G'' means a discrete subgroup Γ of ''G'' such that the manifold ''G''/Γ has finite volume (with respect to a ''G''-invariant measure). For example, a discrete subgroup Γ is a lattice if ''G''/Γ is compact. The [[arithmetic group#Margulis arithmeticity theorem|Margulis arithmeticity theorem]] says, in particular: for a simple Lie group ''G'' of real rank at least 2, every lattice in ''G'' is an arithmetic group.

==The Galois action on the Dynkin diagram==
{{Main article|Tits index}}
In seeking to classify reductive groups which need not be split, one step is the [[Tits index]], which reduces the problem to the case of anisotropic groups. This reduction generalizes several fundamental theorems in algebra. For example, [[Witt's decomposition theorem]] says that a nondegenerate quadratic form over a field is determined up to isomorphism by its Witt index together with its anisotropic kernel. Likewise, the [[Artin–Wedderburn theorem]] reduces the classification of central simple algebras over a field to the case of division algebras. Generalizing these results, Tits showed that a reductive group over a field ''k'' is determined up to isomorphism by its Tits index together with its anisotropic kernel, an associated anisotropic semisimple ''k''-group.

For a reductive group ''G'' over a field ''k'', the [[absolute Galois group]] Gal(''k''<sub>''s''</sub>/''k'') acts (continuously) on the "absolute" Dynkin diagram of ''G'', that is, the Dynkin diagram of ''G'' over a [[separable closure]] ''k''<sub>s</sub> (which is also the Dynkin diagram of ''G'' over an algebraic closure <math>{\overline k}</math>). The Tits index of ''G'' consists of the root datum of ''G''<sub>''k''<sub>''s''</sub></sub>, the Galois action on its Dynkin diagram, and a Galois-invariant subset of the vertices of the Dynkin diagram. Traditionally, the Tits index is drawn by circling the Galois orbits in the given subset.

There is a full classification of quasi-split groups in these terms. Namely, for each action of the absolute Galois group of a field ''k'' on a Dynkin diagram, there is a unique simply connected semisimple quasi-split group ''H'' over ''k'' with the given action. (For a quasi-split group, every Galois orbit in the Dynkin diagram is circled.) Moreover, any other simply connected semisimple group ''G'' over ''k'' with the given action is an [[inner form]] of the quasi-split group ''H'', meaning that ''G'' is the group associated to an element of the [[Galois cohomology]] set ''H''<sup>1</sup>(''k'',''H''/''Z''), where ''Z'' is the center of ''H''. In other words, ''G'' is the twist of ''H'' associated to some ''H''/''Z''-torsor over ''k'', as discussed in the next section.

Example: Let ''q'' be a nondegenerate quadratic form of even dimension 2''n'' over a field ''k'' of characteristic not 2, with ''n'' ≥ 5. (These restrictions can be avoided.) Let ''G'' be the simple group ''SO''(''q'') over ''k''. The absolute Dynkin diagram of ''G'' is of type D<sub>''n''</sub>, and so its automorphism group is of order 2, switching the two "legs" of the D<sub>''n''</sub> diagram. The action of the absolute Galois group of ''k'' on the Dynkin diagram is trivial if and only if the signed [[discriminant of a quadratic form|discriminant]] ''d'' of ''q'' in ''k''*/(''k''*)<sup>2</sup> is trivial. If ''d'' is nontrivial, then it is encoded in the Galois action on the Dynkin diagram: the index-2 subgroup of the Galois group that acts as the identity is <math>\operatorname{Gal}(k_s/k(\sqrt{d}))\subset \operatorname{Gal}(k_s/k)</math>. The group ''G'' is split if and only if ''q'' has Witt index ''n'', the maximum possible, and ''G'' is quasi-split if and only if ''q'' has Witt index at least ''n'' − 1.<ref name = "B234" />

==Torsors and the Hasse principle==
A '''[[torsor]]''' for an affine group scheme ''G'' over a field ''k'' means an affine scheme ''X'' over ''k'' with an [[Group action (mathematics)|action]] of ''G'' such that <math>X_{\overline k}</math> is isomorphic to <math>G_{\overline k}</math> with the action of <math>G_{\overline k}</math> on itself by left translation. A torsor can also be viewed as a [[principal G-bundle]] over ''k'' with respect to the [[fppf topology]] on ''k'', or the [[étale topology]] if ''G'' is smooth over ''k''. The [[pointed set]] of isomorphism classes of ''G''-torsors over ''k'' is called ''H''<sup>1</sup>(''k'',''G''), in the language of Galois cohomology.

Torsors arise whenever one seeks to classify '''forms''' of a given algebraic object ''Y'' over a field ''k'', meaning objects ''X'' over ''k'' which become isomorphic to ''Y'' over the algebraic closure of ''k''. Namely, such forms (up to isomorphism) are in one-to-one correspondence with the set ''H''<sup>1</sup>(''k'',Aut(''Y'')). For example, (nondegenerate) quadratic forms of dimension ''n'' over ''k'' are classified by ''H''<sup>1</sup>(''k'',''O''(''n'')), and central simple algebras of degree ''n'' over ''k'' are classified by ''H''<sup>1</sup>(''k'',''PGL''(''n'')). Also, ''k''-forms of a given algebraic group ''G'' (sometimes called "twists" of ''G'') are classified by ''H''<sup>1</sup>(''k'',Aut(''G'')). These problems motivate the systematic study of ''G''-torsors, especially for reductive groups ''G''.

When possible, one hopes to classify ''G''-torsors using [[cohomological invariant]]s, which are invariants taking values in Galois cohomology with ''abelian'' coefficient groups ''M'', ''H''<sup>''a''</sup>(''k'',''M''). In this direction, Steinberg proved [[Jean-Pierre Serre|Serre]]'s "Conjecture I": for a connected linear algebraic group ''G'' over a perfect field of [[cohomological dimension of a field|cohomological dimension]] at most 1, ''H''<sup>1</sup>(''k'',''G'') = 1.<ref>Steinberg (1965), Theorem 1.9.</ref> (The case of a finite field was known earlier, as [[Lang's theorem]].) It follows, for example, that every reductive group over a finite field is quasi-split.

[[Serre's conjecture II (algebra)|Serre's Conjecture II]] predicts that for a simply connected semisimple group ''G'' over a field of cohomological dimension at most 2, ''H''<sup>1</sup>(''k'',''G'') = 1. The conjecture is known for a [[totally imaginary number field]] (which has cohomological dimension 2). More generally, for any number field ''k'', [[Martin Kneser]], [[Günter Harder]] and Vladimir Chernousov (1989) proved the [[Hasse principle]]: for a simply connected semisimple group ''G'' over ''k'', the map
:<math>H^1(k,G)\to \prod_{v} H^1(k_v,G)</math>
is bijective.<ref>Platonov & Rapinchuk (1994), Theorem 6.6.</ref> Here ''v'' runs over all [[place (mathematics)|places]] of ''k'', and ''k''<sub>''v''</sub> is the corresponding local field (possibly '''R''' or '''C'''). Moreover, the pointed set ''H''<sup>1</sup>(''k''<sub>''v''</sub>,''G'') is trivial for every nonarchimidean local field ''k''<sub>''v''</sub>, and so only the real places of ''k'' matter. The analogous result for a [[global field]] ''k'' of positive characteristic was proved earlier by Harder (1975): for every simply connected semisimple group ''G'' over ''k'', ''H''<sup>1</sup>(''k'',''G'') is trivial (since ''k'' has no real places).<ref>Platonov & Rapinchuk (1994), section 6.8.</ref>

In the slightly different case of an adjoint group ''G'' over a number field ''k'', the Hasse principle holds in a weaker form: the natural map
:<math>H^1(k,G)\to \prod_{v} H^1(k_v,G)</math>
is injective.<ref>Platonov & Rapinchuk (1994), Theorem 6.4.</ref>  For ''G'' = ''PGL''(''n''), this amounts to the [[Albert–Brauer–Hasse–Noether theorem]], saying that a central simple algebra over a number field is determined by its local invariants.

Building on the Hasse principle, the classification of semisimple groups over number fields is well understood. For example, there are exactly three '''Q'''-forms of the exceptional group [[E8 (mathematics)|E<sub>8</sub>]], corresponding to the three real forms of E<sub>8</sub>.

==See also==
*The [[Group of Lie type|groups of Lie type]] are the finite simple groups constructed from simple algebraic groups over finite fields.
*[[Generalized flag variety]], [[Bruhat decomposition]], [[Schubert variety]], [[Schubert calculus]]
*[[Schur algebra]], [[Deligne–Lusztig theory]]
*[[Real form (Lie theory)]]
*[[Weil's conjecture on Tamagawa numbers]]
*[[Langlands classification]], [[Langlands dual group]], [[Langlands program]], [[geometric Langlands program]]
*[[Special group (algebraic group theory)|Special group]], [[essential dimension]]
*[[Geometric invariant theory]], [[Luna's slice theorem]], [[Haboush's theorem]]
*[[Radical of an algebraic group]]

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

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*{{Citation | author1-last=Springer | author1-first=Tonny A. | author1-link=T. A. Springer | chapter=Reductive groups | title=Automorphic Forms, Representations, and ''L''-functions | volume=1 | pages=3–27 | publisher=[[American Mathematical Society]] | isbn=0-8218-3347-2 | mr=0546587 | year=1979 | chapter-url=https://bookstore.ams.org/pspum-33-1}}
*{{Citation | author1-last=Springer | author1-first=Tonny A. | author1-link=T. A. Springer | title=Linear Algebraic Groups | publisher=Birkhäuser Boston | location=Boston, MA | edition=2nd | series=Progress in Mathematics | isbn=978-0-8176-4021-7 |mr=1642713 | year=1998 | volume=9 | doi=10.1007/978-0-8176-4840-4}}
*{{Citation | author1-last=Steinberg | author1-first=Robert | author1-link=Robert Steinberg | title=Regular elements of semisimple algebraic groups | journal=[[Publications Mathématiques de l'IHÉS]] | volume=25 | year=1965 | pages=49–80 | mr=0180554 | url=http://www.numdam.org/item/PMIHES_1965__25__49_0 | doi=10.1007/bf02684397| s2cid=55638217 }}
*{{Citation | author1-last=Steinberg | author1-first=Robert | author1-link=Robert Steinberg | title=Lectures on Chevalley Groups | volume=66 | orig-year=1968 | year=2016 | publisher=[[American Mathematical Society]] | isbn=978-1-4704-3105-1 | mr=3616493 | doi=10.1090/ulect/066| series=University Lecture Series }}
*{{Citation | author1-last=Tits | author1-first=Jacques | author1-link=Jacques Tits | title=Algebraic and abstract simple groups | journal=[[Annals of Mathematics]] | volume=80 | issue=2 | year=1964 | pages=313–329 | mr=0164968 |doi=10.2307/1970394| jstor=1970394 }}

==External links==
*{{Citation | author1-last=Demazure | author1-first=M. | author1-link=Michel Demazure | author2-last=Grothendieck | author2-first=A. | author2-link=Alexander Grothendieck | editor1-last=Gille | editor1-first=P. | editor2-last=Polo | editor2-first=P. | title = Schémas en groupes (SGA 3), II: Groupes de type multiplicatif, et structure des schémas en groupes généraux | url=https://webusers.imj-prg.fr/~patrick.polo/SGA3/}} Revised and annotated edition of the 1970 original.

[[Category:Linear algebraic groups]]
[[Category:Lie groups]]