Inmathematics, areductive group is a type oflinear algebraic group over afield. One definition is that a connected linear algebraic groupG over aperfect field is reductive if it has arepresentation that has a finitekernel and is adirect sum ofirreducible representations. Reductive groups include some of the most important groups in mathematics, such as thegeneral linear groupGL(n) ofinvertible matrices, thespecial orthogonal groupSO(n), and thesymplectic groupSp(2n).Simple algebraic groups and (more generally)semisimple algebraic groups are reductive.

Claude Chevalley showed that the classification of reductive groups is the same over anyalgebraically closed field. In particular, the simple algebraic groups are classified byDynkin diagrams, as in the theory ofcompact Lie groups orcomplexsemisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as thereal numbersR or anumber field, the classification is well understood. Theclassification of finite simple groups says that most finite simple groups arise as the groupG(k) ofk-rational points of a simple algebraic groupG over afinite fieldk, or as minor variants of that construction.

Reductive groups have a richrepresentation theory in various contexts. First, one can study the representations of a reductive groupG over a fieldk as an algebraic group, which are actions ofG onk-vector spaces. But also, one can study the complex representations of the groupG(k) whenk is a finite field, or the infinite-dimensionalunitary representations of a real reductive group, or theautomorphic representations of anadelic algebraic group. The structure theory of reductive groups is used in all these areas.

Alinear algebraic group over a fieldk is defined as asmooth closedsubgroup scheme ofGL(n) overk, for some positive integern. Equivalently, a linear algebraic group overk is a smoothaffine group scheme overk.

Aconnected linear algebraic group${\displaystyle G}$ over an algebraically closed field is calledsemisimple if every smooth connectedsolvablenormal subgroup of${\displaystyle G}$ is trivial. More generally, a connected linear algebraic group${\displaystyle G}$ over an algebraically closed field is calledreductive if the largest smooth connectedunipotent normal subgroup of${\displaystyle G}$ is trivial.^[1] This normal subgroup is called theunipotent radical and is denoted${\displaystyle R_u(G)}$. (Some authors do not require reductive groups to be connected.) A group${\displaystyle G}$ over an arbitrary fieldk is called semisimple or reductive if thebase change${\displaystyle G_{\overline{k}}}$ is semisimple or reductive, where${\displaystyle \overline{k}}$ is analgebraic closure ofk. (This is equivalent to the definition of reductive groups in the introduction whenk is perfect.^[2]) Anytorus overk, such as themultiplicative groupG_m, is reductive.

Over fields of characteristic zero another equivalent definition of a reductive group is a connected group${\displaystyle G}$ admitting a faithful semisimple representation which remains semisimple over its algebraic closure${\displaystyle k^{al}}$^{[3]page 424}.

A linear algebraic groupG over a fieldk is calledsimple (ork-simple) if it is semisimple, nontrivial, and every smooth connected normal subgroup ofG overk is trivial or equal toG.^[4] (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivialcenter (although the center must be finite). For example, for any integern at least 2 and any fieldk, the groupSL(n) overk is simple, and its center is thegroup scheme μ_n ofnth roots of unity.

Acentral isogeny of reductive groups is a surjectivehomomorphism with kernel a finitecentral 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 fieldk,

${\displaystyle GL(n)\cong (G_m\times SL(n))/{\mu }_n.}$

It is slightly awkward that the definition of a reductive group over a field involves passage to the algebraic closure. For a perfect fieldk, that can be avoided: a linear algebraic groupG overk is reductive if and only if every smooth connected unipotent normalk-subgroup ofG is trivial. For an arbitrary field, the latter property defines apseudo-reductive group, which is somewhat more general.

A reductive groupG over a fieldk is calledsplit if it contains a split maximal torusT overk (that is, asplit torus inG whose base change to${\displaystyle \overline{k}}$ is amaximal torus in${\displaystyle G_{\overline{k}}}$). It is equivalent to say thatT is a split torus inG that is maximal among allk-tori inG.^[5] These kinds of groups are useful because their classification can be described through combinatorical data called root data.

A fundamental example of a reductive group is thegeneral linear group${\displaystyle {\text{GL}}_n}$ of invertiblen ×n matrices over a fieldk, for a natural numbern. In particular, themultiplicative groupG_m is the groupGL(1), and so its groupG_m(k) ofk-rational points is the groupk* of nonzero elements ofk under multiplication. Another reductive group is thespecial linear groupSL(n) over a fieldk, the subgroup of matrices withdeterminant 1. In fact,SL(n) is a simple algebraic group forn at least 2.

An important simple group is thesymplectic groupSp(2n) over a fieldk, the subgroup ofGL(2n) that preserves a nondegenerate alternatingbilinear form on thevector spacek²ⁿ. Likewise, theorthogonal groupO(q) is the subgroup of the general linear group that preserves a nondegeneratequadratic formq on a vector space over a fieldk. The algebraic groupO(q) has twoconnected components, and itsidentity componentSO(q) is reductive, in fact simple forq of dimensionn at least 3. (Fork of characteristic 2 andn odd, the group schemeO(q) is in fact connected but not smooth overk. The simple groupSO(q) can always be defined as the maximal smooth connected subgroup ofO(q) overk.) Whenk is algebraically closed, any two (nondegenerate) quadratic forms of the same dimension are isomorphic, and so it is reasonable to call this groupSO(n). For a general fieldk, different quadratic forms of dimensionn can yield non-isomorphic simple groupsSO(q) overk, although they all have the same base change to the algebraic closure${\displaystyle \overline{k}}$.

The group${\displaystyle {\mathbb{G}}_m}$ and products of it are called thealgebraic tori. They are examples of reductive groups since they embed in${\displaystyle {\text{GL}}_n}$ through the diagonal, and from this representation, their unipotent radical is trivial. For example,${\displaystyle {\mathbb{G}}_m\times {\mathbb{G}}_m}$ embeds in${\displaystyle {\text{GL}}_2}$ from the map

• Anyunipotent group is not reductive since its unipotent radical is itself. This includes the additive group${\displaystyle {\mathbb{G}}_a}$.
• TheBorel group${\displaystyle B_n}$ of${\displaystyle {\text{GL}}_n}$ has a non-trivial unipotent radical${\displaystyle {\mathbb{U}}_n}$ of upper-triangular matrices with${\displaystyle 1}$ on the diagonal. This is an example of a non-reductive group which is not unipotent.

Note that the normality of the unipotent radical${\displaystyle R_u(G)}$ implies that the quotient group${\displaystyle G/R_u(G)}$ is reductive. For example,

${\displaystyle B_n/(R_u(B_n))\cong \prod _{i=1}^n{\mathbb{G}}_m.}$

Every compact connected Lie group has acomplexification, 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 groupK with complexificationG, the inclusion fromK into the complex reductive groupG(C) is ahomotopy equivalence, with respect to the classical topology onG(C). For example, the inclusion from theunitary groupU(n) toGL(n,C) is a homotopy equivalence.

For a reductive groupG over a field ofcharacteristic zero, all finite-dimensional representations ofG (as an algebraic group) arecompletely reducible, that is, they are direct sums of irreducible representations.^[6] 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 schemeG offinite type over a fieldk is calledlinearly reductive if its finite-dimensional representations are completely reducible. Fork of characteristic zero,G is linearly reductive if and only if the identity componentG^o ofG is reductive.^[7] Fork of characteristicp>0, however,Masayoshi Nagata showed thatG is linearly reductive if and only ifG^o is ofmultiplicative type andG/G^o has order prime top.^[8]

The classification of reductive algebraic groups is in terms of the associatedroot system, as in the theories of complex semisimple Lie algebras or compact Lie groups. Here is the way roots appear for reductive groups.

LetG be a split reductive group over a fieldk, and letT be a split maximal torus inG; soT is isomorphic to (G_m)ⁿ for somen, withn called therank ofG. Every representation ofT (as an algebraic group) is a direct sum of 1-dimensional representations.^[9] Aweight forG means an isomorphism class of 1-dimensional representations ofT, or equivalently a homomorphismT →G_m. The weights form a groupX(T) undertensor product of representations, withX(T) isomorphic to the product ofn copies of theintegers,Zⁿ.

Theadjoint representation is the action ofG by conjugation on itsLie algebra${\displaystyle \mathfrak{g}}$. Aroot ofG means a nonzero weight that occurs in the action ofT ⊂G on${\displaystyle \mathfrak{g}}$. The subspace of${\displaystyle \mathfrak{g}}$ corresponding to each root is 1-dimensional, and the subspace of${\displaystyle \mathfrak{g}}$ fixed byT is exactly the Lie algebra${\displaystyle \mathfrak{t}}$ ofT.^[10] Therefore, the Lie algebra ofG decomposes into${\displaystyle \mathfrak{t}}$ together with 1-dimensional subspaces indexed by the set Φ of roots:

${\displaystyle \mathfrak{g}=\mathfrak{t}\oplus \underset{\alpha \in \mathrm{\Phi }}{⨁}{\mathfrak{g}}_{\alpha }.}$

For example, whenG is the groupGL(n), its Lie algebra${\displaystyle \mathfrak{g}l(n)}$ is the vector space of alln ×n matrices overk. LetT be the subgroup of diagonal matrices inG. Then the root-space decomposition expresses${\displaystyle \mathfrak{g}l(n)}$ as the direct sum of the diagonal matrices and the 1-dimensional subspaces indexed by the off-diagonal positions (i,j). WritingL₁,...,L_n for the standard basis for the weight latticeX(T) ≅Zⁿ, the roots are the elementsL_i −L_j for alli ≠j from 1 ton.

The roots of a semisimple group form aroot system; this is a combinatorial structure which can be completely classified. More generally, the roots of a reductive group form aroot datum, a slight variation.^[11] TheWeyl group of a reductive groupG means thequotient group of thenormalizer of a maximal torus by the torus,W =N_G(T)/T. The Weyl group is in fact a finite group generated by reflections. For example, for the groupGL(n) (orSL(n)), the Weyl group is thesymmetric groupS_n.

There are finitely manyBorel subgroups containing a given maximal torus, and they are permutedsimply transitively by the Weyl group (acting byconjugation).^[12] A choice of Borel subgroup determines a set ofpositive roots Φ⁺ ⊂ Φ, with the property that Φ is the disjoint union of Φ⁺ and −Φ⁺. Explicitly, the Lie algebra ofB is the direct sum of the Lie algebra ofT and the positive root spaces:

${\displaystyle \mathfrak{b}=\mathfrak{t}\oplus \underset{\alpha \in {\mathrm{\Phi }}^{+}}{⨁}{\mathfrak{g}}_{\alpha }.}$

For example, ifB is the Borel subgroup of upper-triangular matrices inGL(n), then this is the obvious decomposition of the subspace${\displaystyle \mathfrak{b}}$ of upper-triangular matrices in${\displaystyle \mathfrak{g}l(n)}$. The positive roots areL_i −L_j for 1 ≤i <j ≤n.

Asimple root means a positive root that is not a sum of two other positive roots. Write Δ for the set of simple roots. The numberr of simple roots is equal to the rank of thecommutator subgroup ofG, called thesemisimple rank ofG (which is simply the rank ofG ifG is semisimple). For example, the simple roots forGL(n) (orSL(n)) areL_i −L_i+1 for 1 ≤i ≤n − 1.

Root systems are classified by the correspondingDynkin diagram, which is a finitegraph (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-invariantinner product on the weight lattice. The connected Dynkin diagrams (corresponding to simple groups) are pictured below.

For a split reductive groupG over a fieldk, an important point is that a root α determines not just a 1-dimensional subspace of the Lie algebra ofG, but also a copy of the additive groupG_a inG with the given Lie algebra, called aroot subgroupU_α. The root subgroup is the unique copy of the additive group inG which isnormalized byT and which has the given Lie algebra.^[10] The whole groupG is generated (as an algebraic group) byT and the root subgroups, while the Borel subgroupB is generated byT and the positive root subgroups. In fact, a split semisimple groupG is generated by the root subgroups alone.

For a split reductive groupG over a fieldk, the smooth connected subgroups ofG that contain a given Borel subgroupB ofG 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). Letr be the order of Δ, the semisimple rank ofG. Everyparabolic subgroup ofG isconjugate to a subgroup containingB by some element ofG(k). As a result, there are exactly 2^r conjugacy classes of parabolic subgroups inG overk.^[13] Explicitly, the parabolic subgroup corresponding to a given subsetS of Δ is the group generated byB together with the root subgroupsU_−α for α inS. For example, the parabolic subgroups ofGL(n) that contain the Borel subgroupB above are the groups of invertible matrices with zero entries below a given set of squares along the diagonal, such as:

By definition, aparabolic subgroupP of a reductive groupG over a fieldk is a smoothk-subgroup such that the quotient varietyG/P isproper overk, or equivalentlyprojective overk. Thus the classification of parabolic subgroups amounts to a classification of theprojective homogeneous varieties forG (with smooth stabilizer group; that is no restriction fork of characteristic zero). ForGL(n), these are theflag varieties, parametrizing sequences of linear subspaces of given dimensionsa₁,...,a_i contained in a fixed vector spaceV of dimensionn:

${\displaystyle 0\subset S_{a_1}\subset \cdots \subset S_{a_i}\subset V.}$

For the orthogonal group or the symplectic group, the projective homogeneous varieties have a similar description as varieties ofisotropic flags with respect to a given quadratic form or symplectic form. For any reductive groupG with a Borel subgroupB,G/B is called theflag variety orflag manifold ofG.

Chevalley showed in 1958 that the reductive groups over any algebraically closed field are classified up to isomorphism by root data.^[14] 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_n, B_n, C_n, D_n, E₆, E₇, E₈, F₄, G₂. This result is essentially identical to the classifications of compact Lie groups or complex semisimple Lie algebras, byWilhelm Killing andÉlie Cartan in the 1880s and 1890s. In particular, the dimensions, centers, and other properties of the simple algebraic groups can be read from thelist 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.

Theexceptional groupsG of type G₂ and E₆ had been constructed earlier, at least in the form of the abstract groupG(k), byL. E. Dickson. For example, the groupG₂ is theautomorphism group of anoctonion algebra overk. By contrast, the Chevalley groups of type F₄, E₇, E₈ over a field of positive characteristic were completely new.

More generally, the classification ofsplit reductive groups is the same over any field.^[15] A semisimple groupG over a fieldk is calledsimply connected if every central isogeny from a semisimple group toG is an isomorphism. (ForG semisimple over thecomplex numbers, being simply connected in this sense is equivalent toG(C) beingsimply connected in the classical topology.) Chevalley's classification gives that, over any fieldk, there is a unique simply connected split semisimple groupG with a given Dynkin diagram, with simple groups corresponding to the connected diagrams. At the other extreme, a semisimple group is ofadjoint type if its center is trivial. The split semisimple groups overk with given Dynkin diagram are exactly the groupsG/A, whereG is the simply connected group andA is ak-subgroup scheme of the center ofG.

For example, the simply connected split simple groups over a fieldk corresponding to the "classical" Dynkin diagrams are as follows:

• A_n:SL(n+1) overk;
• B_n: thespin group Spin(2n+1) associated to a quadratic form of dimension 2n+1 overk withWitt indexn, for example the form

${\displaystyle q(x_1,\dots ,x_{2n+1})=x_1{x}_2+x_3{x}_4+\cdots +x_{2n-1}{x}_{2n}+x_{2n+1}^2;}$

• C_n: the symplectic groupSp(2n) overk;
• D_n: the spin group Spin(2n) associated to a quadratic form of dimension 2n overk with Witt indexn, which can be written as:

${\displaystyle q(x_1,\dots ,x_{2n})=x_1{x}_2+x_3{x}_4+\cdots +x_{2n-1}{x}_{2n}.}$

Theouter automorphism group of a split reductive groupG over a fieldk is isomorphic to the automorphism group of the root datum ofG. Moreover, the automorphism group ofG splits as asemidirect product:

${\displaystyle \mathrm{Aut}(G)\cong \mathrm{Out}(G)⋉(G/Z)(k),}$

whereZ is the center ofG.^[16] For a split semisimple simply connected groupG over a field, the outer automorphism group ofG has a simpler description: it is the automorphism group of the Dynkin diagram ofG.

Agroup schemeG over a schemeS is calledreductive if the morphismG →S issmooth and affine, and every geometric fiber${\displaystyle G_{\overline{k}}}$ is reductive. (For a pointp inS, the corresponding geometric fiber means the base change ofG to an algebraic closure${\displaystyle \overline{k}}$ of the residue field ofp.) Extending Chevalley's work,Michel Demazure and Grothendieck showed thatpinned reductive group schemes over any nonempty schemeS are classified by root data.^[17] This statement includes the existence of Chevalley groups as group schemes overZ, and it says that every pinned reductive group over a schemeS is isomorphic to the base change of a Chevalley group fromZ toS. 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 thatA is a Dedekind domain and thatI is an ideal inA whose class in the class group ofA is not a square. ThenSL(A + I) and SL_2(A) are split and reductive overSpec 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 bundleP(A + I) and has no section with trivial normal bundle (a section corresponds to a short exact sequence0 → J → A + I → K → 0 whereJ, K are ideal classes and the normal bundle is thenJ^{-1}K, which is not trivial sinceJK is isomorphic toI) while the flag scheme of the second isP^1_A and does possess sections with trivial normal bundle.

In the context ofLie groups rather than algebraic groups, areal reductive group is a Lie groupG such that there is a linear algebraic groupL overR whose identity component (in theZariski topology) is reductive, and a homomorphismG →L(R) whose kernel is finite and whose image is open inL(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_C) = Ad(L⁰(C)) (which is automatic forG connected).^[18]

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

Useful theories ofadmissible representations 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 groupG overR may be connected as an algebraic group while the Lie groupG(R) is not connected, and likewise for simply connected groups.

For example, theprojective linear groupPGL(2) is connected as an algebraic group over any field, but its group of real pointsPGL(2,R) has two connected components. The identity component ofPGL(2,R) (sometimes calledPSL(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 groupSL(2,R) hasfundamental group isomorphic to the integersZ, and soSL(2,R) has nontrivialcovering spaces. By definition, all finite coverings ofSL(2,R) (such as themetaplectic group) are real reductive groups. On the other hand, theuniversal cover ofSL(2,R) is not a real reductive group, even though its Lie algebra isreductive, that is, the product of a semisimple Lie algebra and an abelian Lie algebra.

For a connected real reductive groupG, the quotient manifoldG/K ofG by amaximal compact subgroupK is asymmetric space of non-compact type. In fact, every symmetric space of non-compact type arises this way. These are central examples inRiemannian geometry of manifolds with nonpositivesectional curvature. For example,SL(2,R)/SO(2) is thehyperbolic plane, andSL(2,C)/SU(2) is hyperbolic 3-space.

For a reductive groupG over a fieldk that is complete with respect to adiscrete valuation (such as thep-adic numbersQ_p), theaffine buildingX ofG plays the role of the symmetric space. Namely,X is asimplicial complex with an action ofG(k), andG(k) preserves aCAT(0) metric onX, the analog of a metric with nonpositive curvature. The dimension of the affine building is thek-rank ofG. For example, the building ofSL(2,Q_p) is atree.

For a split reductive groupG over a fieldk, the irreducible representations ofG (as an algebraic group) are parametrized by thedominant weights, which are defined as the intersection of the weight latticeX(T) ≅Zⁿ with a convex cone (aWeyl chamber) inRⁿ. In particular, this parametrization is independent of the characteristic ofk. In more detail, fix a split maximal torus and a Borel subgroup,T ⊂B ⊂G. ThenB is the semidirect product ofT with a smooth connected unipotent subgroupU. Define ahighest weight vector in a representationV ofG overk to be a nonzero vectorv such thatB maps the line spanned byv into itself. ThenB acts on that line through its quotient groupT, by some element λ of the weight latticeX(T). Chevalley showed that every irreducible representation ofG 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 representationL(λ) ofG, up to isomorphism.^[19]

There remains the problem of describing the irreducible representation with given highest weight. Fork of characteristic zero, there are essentially complete answers. For a dominant weight λ, define theSchur module ∇(λ) as thek-vector space of sections of theG-equivariantline bundle on the flag manifoldG/B associated to λ; this is a representation ofG. Fork of characteristic zero, theBorel–Weil theorem says that the irreducible representationL(λ) is isomorphic to the Schur module ∇(λ). Furthermore, theWeyl character formula gives thecharacter (and in particular the dimension) of this representation.

For a split reductive groupG over a fieldk of positive characteristic, the situation is far more subtle, because representations ofG are typically not direct sums of irreducibles. For a dominant weight λ, the irreducible representationL(λ) is the unique simple submodule (thesocle) 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), byGeorge Kempf.^[20] The dimensions and characters of the irreducible representationsL(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 ofL(λ) are known when the characteristicp ofk is much bigger than theCoxeter number ofG, byHenning Andersen,Jens Jantzen, and Wolfgang Soergel (provingLusztig's conjecture in that case). Their character formula forp large is based on theKazhdan–Lusztig polynomials, which are combinatorially complex.^[21] For any primep, Simon Riche andGeordie Williamson conjectured the irreducible characters of a reductive group in terms of thep-Kazhdan-Lusztig polynomials, which are even more complex, but at least are computable.^[22]

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 theclassical groups are:

• Every nondegenerate quadratic formq over a fieldk determines a reductive group G =SO(q). HereG is simple ifq has dimensionn at least 3, since${\displaystyle G_{\overline{k}}}$ is isomorphic toSO(n) over an algebraic closure${\displaystyle \overline{k}}$. Thek-rank ofG is equal to theWitt index ofq (the maximum dimension of an isotropic subspace overk).^[23] So the simple groupG is split overk if and only ifq has the maximum possible Witt index,${\displaystyle \lfloor n/2\rfloor }$.
• Everycentral simple algebraA overk determines a reductive groupG =SL(1,A), the kernel of thereduced norm on thegroup of unitsA* (as an algebraic group overk). Thedegree ofA means the square root of the dimension ofA as ak-vector space. HereG is simple ifA has degreen at least 2, since${\displaystyle G_{\overline{k}}}$ is isomorphic toSL(n) over${\displaystyle \overline{k}}$. IfA has indexr (meaning thatA is isomorphic to the matrix algebraM_n/r(D) for adivision algebraD of degreer overk), then thek-rank ofG is (n/r) − 1.^[24] So the simple groupG is split overk if and only ifA is a matrix algebra overk.

As a result, the problem of classifying reductive groups overk essentially includes the problem of classifying all quadratic forms overk or all central simple algebras overk. These problems are easy fork 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 fieldk is calledisotropic if it hask-rank greater than 0 (that is, if it contains a nontrivial split torus), and otherwiseanisotropic. For a semisimple groupG over a fieldk, the following conditions are equivalent:

• G is isotropic (that is,G contains a copy of the multiplicative groupG_m overk);
• G contains a parabolic subgroup overk not equal toG;
• G contains a copy of the additive groupG_a overk.

Fork perfect, it is also equivalent to say thatG(k) contains aunipotent element other than 1.^[25]

For a connected linear algebraic groupG over a local fieldk of characteristic zero (such as the real numbers), the groupG(k) iscompact in the classical topology (based on the topology ofk) if and only ifG is reductive and anisotropic.^[26] Example: the orthogonal groupSO(p,q) overR has real rank min(p,q), and so it is anisotropic if and only ifp orq is zero.^[23]

A reductive groupG over a fieldk is calledquasi-split if it contains a Borel subgroup overk. A split reductive group is quasi-split. IfG is quasi-split overk, then any two Borel subgroups ofG are conjugate by some element ofG(k).^[27] Example: the orthogonal groupSO(p,q) overR is split if and only if |p−q| ≤ 1, and it is quasi-split if and only if |p−q| ≤ 2.^[23]

For a simply connected split semisimple groupG over a fieldk,Robert Steinberg gave an explicitpresentation of the abstract groupG(k).^[28] It is generated by copies of the additive group ofk indexed by the roots ofG (the root subgroups), with relations determined by the Dynkin diagram ofG.

For a simply connected split semisimple groupG over a perfect fieldk, Steinberg also determined the automorphism group of the abstract groupG(k). Every automorphism is the product of aninner automorphism, a diagonal automorphism (meaning conjugation by a suitable${\displaystyle \overline{k}}$-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 fieldk).^[29]

For ak-simple algebraic groupG,Tits's simplicity theorem says that the abstract groupG(k) is close to being simple, under mild assumptions. Namely, suppose thatG is isotropic overk, and suppose that the fieldk has at least 4 elements. LetG(k)⁺ be the subgroup of the abstract groupG(k) generated byk-points of copies of the additive groupG_a overk contained inG. (By the assumption thatG is isotropic overk, the groupG(k)⁺ is nontrivial, and even Zariski dense inG ifk is infinite.) Then the quotient group ofG(k)⁺ by its center is simple (as an abstract group).^[30] The proof usesJacques Tits's machinery ofBN-pairs.

The exceptions for fields of order 2 or 3 are well understood. Fork =F₂, Tits's simplicity theorem remains valid except whenG is split of typeA₁,B₂, orG₂, or non-split (that is, unitary) of typeA₂. Fork =F₃, the theorem holds except forG of typeA₁.^[31]

For ak-simple groupG, in order to understand the whole groupG(k), one can consider theWhitehead groupW(k,G)=G(k)/G(k)⁺. ForG simply connected and quasi-split, the Whitehead group is trivial, and so the whole groupG(k) is simple modulo its center.^[32] More generally, theKneser–Tits problem asks for which isotropick-simple groups the Whitehead group is trivial. In all known examples,W(k,G) is abelian.

For an anisotropick-simple groupG, the abstract groupG(k) can be far from simple. For example, letD be a division algebra with center ap-adic fieldk. Suppose that the dimension ofD overk is finite and greater than 1. ThenG =SL(1,D) is an anisotropick-simple group. As mentioned above,G(k) is compact in the classical topology. Since it is alsototally disconnected,G(k) is aprofinite group (but not finite). As a result,G(k) contains infinitely many normal subgroups of finiteindex.^[33]

LetG be a linear algebraic group over therational numbersQ. ThenG can be extended to an affine group schemeG overZ, and this determines an abstract groupG(Z). Anarithmetic group means any subgroup ofG(Q) that iscommensurable withG(Z). (Arithmeticity of a subgroup ofG(Q) is independent of the choice ofZ-structure.) For example,SL(n,Z) is an arithmetic subgroup ofSL(n,Q).

For a Lie groupG, alattice inG means a discrete subgroup Γ ofG such that the manifoldG/Γ has finite volume (with respect to aG-invariant measure). For example, a discrete subgroup Γ is a lattice ifG/Γ is compact. TheMargulis arithmeticity theorem says, in particular: for a simple Lie groupG of real rank at least 2, every lattice inG is an arithmetic group.

In seeking to classify reductive groups which need not be split, one step is theTits 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, theArtin–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 fieldk is determined up to isomorphism by its Tits index together with its anisotropic kernel, an associated anisotropic semisimplek-group.

For a reductive groupG over a fieldk, theabsolute Galois group Gal(k_s/k) acts (continuously) on the "absolute" Dynkin diagram ofG, that is, the Dynkin diagram ofG over aseparable closurek_s (which is also the Dynkin diagram ofG over an algebraic closure${\displaystyle \overline{k}}$). The Tits index ofG consists of the root datum ofG_ks, 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 fieldk on a Dynkin diagram, there is a unique simply connected semisimple quasi-split groupH overk with the given action. (For a quasi-split group, every Galois orbit in the Dynkin diagram is circled.) Moreover, any other simply connected semisimple groupG overk with the given action is aninner form of the quasi-split groupH, meaning thatG is the group associated to an element of theGalois cohomology setH¹(k,H/Z), whereZ is the center ofH. In other words,G is the twist ofH associated to someH/Z-torsor overk, as discussed in the next section.

Example: Letq be a nondegenerate quadratic form of even dimension 2n over a fieldk of characteristic not 2, withn ≥ 5. (These restrictions can be avoided.) LetG be the simple groupSO(q) overk. The absolute Dynkin diagram ofG is of type D_n, and so its automorphism group is of order 2, switching the two "legs" of the D_n diagram. The action of the absolute Galois group ofk on the Dynkin diagram is trivial if and only if the signeddiscriminantd ofq ink*/(k*)² is trivial. Ifd 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${\displaystyle \mathrm{Gal}(k_s/k(\sqrt{d}))\subset \mathrm{Gal}(k_s/k)}$. The groupG is split if and only ifq has Witt indexn, the maximum possible, andG is quasi-split if and only ifq has Witt index at leastn − 1.^[23]

Atorsor for an affine group schemeG over a fieldk means an affine schemeX overk with anaction ofG such that${\displaystyle X_{\overline{k}}}$ is isomorphic to${\displaystyle G_{\overline{k}}}$ with the action of${\displaystyle G_{\overline{k}}}$ on itself by left translation. A torsor can also be viewed as aprincipal G-bundle overk with respect to thefppf topology onk, or theétale topology ifG is smooth overk. Thepointed set of isomorphism classes ofG-torsors overk is calledH¹(k,G), in the language of Galois cohomology.

Torsors arise whenever one seeks to classifyforms of a given algebraic objectY over a fieldk, meaning objectsX overk which become isomorphic toY over the algebraic closure ofk. Namely, such forms (up to isomorphism) are in one-to-one correspondence with the setH¹(k,Aut(Y)). For example, (nondegenerate) quadratic forms of dimensionn overk are classified byH¹(k,O(n)), and central simple algebras of degreen overk are classified byH¹(k,PGL(n)). Also,k-forms of a given algebraic groupG (sometimes called "twists" ofG) are classified byH¹(k,Aut(G)). These problems motivate the systematic study ofG-torsors, especially for reductive groupsG.

When possible, one hopes to classifyG-torsors usingcohomological invariants, which are invariants taking values in Galois cohomology withabelian coefficient groupsM,H^a(k,M). In this direction, Steinberg provedSerre's "Conjecture I": for a connected linear algebraic groupG over a perfect field ofcohomological dimension at most 1,H¹(k,G) = 1.^[34] (The case of a finite field was known earlier, asLang's theorem.) It follows, for example, that every reductive group over a finite field is quasi-split.

Serre's Conjecture II predicts that for a simply connected semisimple groupG over a field of cohomological dimension at most 2,H¹(k,G) = 1. The conjecture is known for atotally imaginary number field (which has cohomological dimension 2). More generally, for any number fieldk,Martin Kneser,Günter Harder and Vladimir Chernousov (1989) proved theHasse principle: for a simply connected semisimple groupG overk, the map

${\displaystyle H^1(k,G)\to \prod _v{H}^1(k_v,G)}$

is bijective.^[35] Herev runs over allplaces ofk, andk_v is the corresponding local field (possiblyR orC). Moreover, the pointed setH¹(k_v,G) is trivial for every nonarchimidean local fieldk_v, and so only the real places ofk matter. The analogous result for aglobal fieldk of positive characteristic was proved earlier by Harder (1975): for every simply connected semisimple groupG overk,H¹(k,G) is trivial (sincek has no real places).^[36]

In the slightly different case of an adjoint groupG over a number fieldk, the Hasse principle holds in a weaker form: the natural map

${\displaystyle H^1(k,G)\to \prod _v{H}^1(k_v,G)}$

is injective.^[37] ForG =PGL(n), this amounts to theAlbert–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 threeQ-forms of the exceptional groupE₈, corresponding to the three real forms of E₈.

• Thegroups 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,essential dimension
• Geometric invariant theory,Luna's slice theorem,Haboush's theorem
• Radical of an algebraic group

• Borel, Armand (1991) [1969],Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), New York:Springer Nature,doi:10.1007/978-1-4612-0941-6,ISBN 0-387-97370-2,MR 1102012
• Borel, Armand;Tits, Jacques (1971), "Éléments unipotents et sous-groupes paraboliques de groupes réductifs. I.",Inventiones Mathematicae,12 (2):95–104,Bibcode:1971InMat..12...95B,doi:10.1007/BF01404653,MR 0294349,S2CID 119837998
• Chevalley, Claude (2005) [1958],Cartier, P. (ed.),Classification des groupes algébriques semi-simples, Collected Works, Vol. 3,Springer Nature,ISBN 3-540-23031-9,MR 2124841
• Conrad, Brian (2014),"Reductive group schemes"(PDF),Autour des schémas en groupes, vol. 1, Paris:Société Mathématique de France, pp. 93–444,ISBN 978-2-85629-794-0,MR 3309122
• Demazure, Michel;Gabriel, Pierre (1970),Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Paris: Masson,ISBN 978-2225616662,MR 0302656
• Demazure, M.;Grothendieck, A. (2011) [1970]. Gille, P.; Polo, P. (eds.).Schémas en groupes (SGA 3), I: Propriétés générales des schémas en groupes.Société Mathématique de France.ISBN 978-2-85629-323-2.MR 2867621. Revised and annotated edition of the 1970 original.
• Demazure, M.;Grothendieck, A. (1970).Groupes de Type Multiplicatif, et Structure des Schémas en Groupes Généraux. Lecture Notes in Mathematics. Vol. 152. Berlin; New York:Springer-Verlag.doi:10.1007/BFb0059005.ISBN 978-3540051800.MR 0274459.
• Demazure, M.;Grothendieck, A. (2011) [1970]. Gille, P.; Polo, P. (eds.).Schémas en groupes (SGA 3), III: Structure des schémas en groupes réductifs.Société Mathématique de France.ISBN 978-2-85629-324-9.MR 2867622. Revised and annotated edition of the 1970 original.
• Gille, Philippe (2009),"Le problème de Kneser–Tits"(PDF),Séminaire Bourbaki. Vol. 2007/2008, Astérisque, vol. 326,Société Mathématique de France, pp. 39–81,ISBN 978-285629-269-3,MR 2605318
• Jantzen, Jens Carsten (2003) [1987],Representations of Algebraic Groups (2nd ed.),American Mathematical Society,ISBN 978-0-8218-3527-2,MR 2015057
• Milne, J. S. (2017),Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field,Cambridge University Press,doi:10.1017/9781316711736,ISBN 978-1107167483,MR 3729270
• Platonov, Vladimir; Rapinchuk, Andrei (1994),Algebraic Groups and Number Theory,Academic Press,ISBN 0-12-558180-7,MR 1278263
• V.L. Popov (2001) [1994],"Reductive group",Encyclopedia of Mathematics,EMS Press
• Riche, Simon; Williamson, Geordie (2018),Tilting Modules and thep-Canonical Basis, Astérisque, vol. 397,Société Mathématique de France,arXiv:1512.08296,Bibcode:2015arXiv151208296R,ISBN 978-2-85629-880-0
• Springer, Tonny A. (1979),"Reductive groups",Automorphic Forms, Representations, andL-functions, vol. 1,American Mathematical Society, pp. 3–27,ISBN 0-8218-3347-2,MR 0546587
• Springer, Tonny A. (1998),Linear Algebraic Groups, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston,doi:10.1007/978-0-8176-4840-4,ISBN 978-0-8176-4021-7,MR 1642713
• Steinberg, Robert (1965),"Regular elements of semisimple algebraic groups",Publications Mathématiques de l'IHÉS,25:49–80,doi:10.1007/bf02684397,MR 0180554,S2CID 55638217
• Steinberg, Robert (2016) [1968],Lectures on Chevalley Groups, University Lecture Series, vol. 66,American Mathematical Society,doi:10.1090/ulect/066,ISBN 978-1-4704-3105-1,MR 3616493
• Tits, Jacques (1964), "Algebraic and abstract simple groups",Annals of Mathematics,80 (2):313–329,doi:10.2307/1970394,JSTOR 1970394,MR 0164968

• Demazure, M.;Grothendieck, A., Gille, P.; Polo, P. (eds.),Schémas en groupes (SGA 3), II: Groupes de type multiplicatif, et structure des schémas en groupes généraux Revised and annotated edition of the 1970 original.

• Linear algebraic groups
• Lie groups

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