Reductive group

Inmathematics,areductive groupis a type oflinear algebraic groupover afield.One definition is that a connected linear algebraic groupGover aperfect fieldis reductive if it has arepresentationthat has a finitekerneland is adirect sumofirreducible 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 groupsand (more generally)semisimple algebraic groupsare reductive.

Claude Chevalleyshowed 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 groupsorcomplexsemisimple Lie algebras.Reductive groups over an arbitrary field are harder to classify, but for many fields such as thereal numbersRor anumber field,the classification is well understood. Theclassification of finite simple groupssays that most finite simple groups arise as the groupG(k) ofk-rational pointsof a simple algebraic groupGover afinite fieldk,or as minor variants of that construction.

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

Definitions

Main article:Linear algebraic group

Alinear algebraic groupover a fieldkis defined as asmoothclosedsubgroup schemeofGL(n) overk,for some positive integern.Equivalently, a linear algebraic group overkis a smoothaffinegroup scheme overk.

With the unipotent radical

Aconnectedlinear algebraic group$G$over an algebraically closed field is calledsemisimpleif every smooth connectedsolvablenormal subgroupof$G$is trivial. More generally, a connected linear algebraic group$G$over an algebraically closed field is calledreductiveif the largest smooth connectedunipotentnormal subgroup of$G$is trivial.This normal subgroup is called theunipotent radicaland is denoted$R_{u}(G)$.(Some authors do not require reductive groups to be connected.) A group$G$over an arbitrary fieldkis called semisimple or reductive if thebase change$G_{\overline {k}}$is semisimple or reductive, where${\overline {k}}$is analgebraic closureofk.(This is equivalent to the definition of reductive groups in the introduction whenkis perfect.) Anytorusoverk,such as themultiplicative groupG_m,is reductive.

With representation theory

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

Simple reductive groups

A linear algebraic groupGover a fieldkis calledsimple(ork-simple) if it is semisimple, nontrivial, and every smooth connected normal subgroup ofGoverkis trivial or equal toG.(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 integernat least 2 and any fieldk,the groupSL(n) overkis simple, and its center is thegroup scheme μ_nofnth roots of unity.

Acentral isogenyof reductive groups is a surjectivehomomorphismwith kernel a finitecentral subgroupscheme. 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,

$GL(n)\cong (G_{m}\times SL(n))/\mu _{n}.$

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 groupGoverkis reductive if and only if every smooth connected unipotent normalk-subgroup ofGis trivial. For an arbitrary field, the latter property defines apseudo-reductive group,which is somewhat more general.

Split-reductive groups

A reductive groupGover a fieldkis calledsplitif it contains a split maximal torusToverk(that is, asplit torusinGwhose base change to${\overline {k}}$is a maximal torus in$G_{\overline {k}}$). It is equivalent to say thatTis a split torus inGthat is maximal among allk-tori inG.These kinds of groups are useful because their classification can be described through combinatorical data called root data.

Examples

GL_nand SL_n

A fundamental example of a reductive group is thegeneral linear group${\text{GL}}_{n}$of invertiblen×nmatrices over a fieldk,for a natural numbern.In particular, themultiplicative groupG_mis the groupGL(1), and so its groupG_m(k) ofk-rational points is the groupk* of nonzero elements ofkunder multiplication. Another reductive group is thespecial linear groupSL(n) over a fieldk,the subgroup of matrices withdeterminant1. In fact,SL(n) is a simple algebraic group fornat least 2.

O(n), SO(n), and Sp(n)

An important simple group is thesymplectic groupSp(2n) over a fieldk,the subgroup ofGL(2n) that preserves a nondegenerate alternatingbilinear formon thevector spacek²ⁿ.Likewise, theorthogonal groupO(q) is the subgroup of the general linear group that preserves a nondegeneratequadratic formqon a vector space over a fieldk.The algebraic groupO(q) has twoconnected components,and itsidentity componentSO(q) is reductive, in fact simple forqof dimensionnat least 3. (Forkof characteristic 2 andnodd, 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.) Whenkis 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 dimensionncan yield non-isomorphic simple groupsSO(q) overk,although they all have the same base change to the algebraic closure${\overline {k}}$.

Tori

The group$\mathbb {G} _{m}$and products of it are called thealgebraic tori.They are examples of reductive groups since they embed in${\text{GL}}_{n}$through the diagonal, and from this representation, their unipotent radical is trivial. For example,$\mathbb {G} _{m}\times \mathbb {G} _{m}$embeds in${\text{GL}}_{2}$from the map

$(a_{1},a_{2})\mapsto {\begin{bmatrix}a_{1}&0\\0&a_{2}\end{bmatrix}}.$

Non-examples

• Anyunipotent groupis not reductive since its unipotent radical is itself. This includes the additive group$\mathbb {G} _{a}$.
• TheBorel group$B_{n}$of${\text{GL}}_{n}$has a non-trivial unipotent radical$\mathbb {U} _{n}$of upper-triangular matrices with$1$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$R_{u}(G)$implies that the quotient group$G/R_{u}(G)$is reductive. For example,

$B_{n}/(R_{u}(B_{n}))\cong \prod _{i=1}^{n}\mathbb {G} _{m}.$

Other characterizations of reductive groups

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 groupKwith complexificationG,the inclusion fromKinto 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 groupGover a field ofcharacteristiczero, all finite-dimensional representations ofG(as an algebraic group) arecompletely reducible,that is, they are direct sums of irreducible representations.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 schemeGoffinite typeover a fieldkis calledlinearly reductiveif its finite-dimensional representations are completely reducible. Forkof characteristic zero,Gis linearly reductive if and only if the identity componentG^oofGis reductive.Forkof characteristicp>0, however,Masayoshi Nagatashowed thatGis linearly reductive if and only ifG^ois ofmultiplicative typeandG/G^ohas order prime top.

Roots

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.

LetGbe a split reductive group over a fieldk,and letTbe a split maximal torus inG;soTis isomorphic to (G_m)ⁿfor somen,withncalled therankofG.Every representation ofT(as an algebraic group) is a direct sum of 1-dimensional representations.AweightforGmeans an isomorphism class of 1-dimensional representations ofT,or equivalently a homomorphismT→G_m.The weights form a groupX(T) undertensor productof representations, withX(T) isomorphic to the product ofncopies of theintegers,Zⁿ.

Theadjoint representationis the action ofGby conjugation on itsLie algebra${\mathfrak {g}}$.ArootofGmeans a nonzero weight that occurs in the action ofT⊂Gon${\mathfrak {g}}$.The subspace of${\mathfrak {g}}$corresponding to each root is 1-dimensional, and the subspace of${\mathfrak {g}}$fixed byTis exactly the Lie algebra${\mathfrak {t}}$ofT.Therefore, the Lie algebra ofGdecomposes into${\mathfrak {t}}$together with 1-dimensional subspaces indexed by the set Φ of roots:

${\mathfrak {g}}={\mathfrak {t}}\oplus \bigoplus _{\ Alpha \in \Phi }{\mathfrak {g}}_{\ Alpha }.$

For example, whenGis the groupGL(n), its Lie algebra${{\mathfrak {g}}l}(n)$is the vector space of alln×nmatrices overk.LetTbe the subgroup of diagonal matrices inG.Then the root-space decomposition expresses${{\mathfrak {g}}l}(n)$as the direct sum of the diagonal matrices and the 1-dimensional subspaces indexed by the off-diagonal positions (i,j). WritingL₁,...,L_nfor the standard basis for the weight latticeX(T) ≅Zⁿ,the roots are the elementsL_i−L_jfor alli≠jfrom 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.TheWeyl groupof a reductive groupGmeans thequotient groupof thenormalizerof 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 subgroupscontaining a given maximal torus, and they are permutedsimply transitivelyby the Weyl group (acting byconjugation).A choice of Borel subgroup determines a set ofpositive rootsΦ⁺⊂ Φ, with the property that Φ is the disjoint union of Φ⁺and −Φ⁺.Explicitly, the Lie algebra ofBis the direct sum of the Lie algebra ofTand the positive root spaces:

${\mathfrak {b}}={\mathfrak {t}}\oplus \bigoplus _{\ Alpha \in \Phi ^{+}}{\mathfrak {g}}_{\ Alpha }.$

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

Asimple rootmeans a positive root that is not a sum of two other positive roots. Write Δ for the set of simple roots. The numberrof simple roots is equal to the rank of thecommutator subgroupofG,called thesemisimple rankofG(which is simply the rank ofGifGis semisimple). For example, the simple roots forGL(n) (orSL(n)) areL_i−L_i+1for 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 producton the weight lattice. The connected Dynkin diagrams (corresponding to simple groups) are pictured below.

For a split reductive groupGover 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_ainGwith the given Lie algebra, called aroot subgroupU_α.The root subgroup is the unique copy of the additive group inGwhich isnormalizedbyTand which has the given Lie algebra.The whole groupGis generated (as an algebraic group) byTand the root subgroups, while the Borel subgroupBis generated byTand the positive root subgroups. In fact, a split semisimple groupGis generated by the root subgroups alone.

Parabolic subgroups

For a split reductive groupGover a fieldk,the smooth connected subgroups ofGthat contain a given Borel subgroupBofGare 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). Letrbe the order of Δ, the semisimple rank ofG.Everyparabolic subgroupofGisconjugateto a subgroup containingBby some element ofG(k). As a result, there are exactly 2^rconjugacy classes of parabolic subgroups inGoverk.Explicitly, the parabolic subgroup corresponding to a given subsetSof Δ is the group generated byBtogether with the root subgroupsU_−αfor α inS.For example, the parabolic subgroups ofGL(n) that contain the Borel subgroupBabove are the groups of invertible matrices with zero entries below a given set of squares along the diagonal, such as:

$\left\{{\begin{bmatrix}*&*&*&*\\*&*&*&*\\0&0&*&*\\0&0&0&*\end{bmatrix}}\right\}$

By definition, aparabolic subgroupPof a reductive groupGover a fieldkis a smoothk-subgroup such that the quotient varietyG/Pisproperoverk,or equivalentlyprojectiveoverk.Thus the classification of parabolic subgroups amounts to a classification of theprojective homogeneous varietiesforG(with smooth stabilizer group; that is no restriction forkof characteristic zero). ForGL(n), these are theflag varieties,parametrizing sequences of linear subspaces of given dimensionsa₁,...,a_icontained in a fixed vector spaceVof dimensionn:

$0\subset S_{a_{1}}\subset \cdots \subset S_{a_{i}}\subset V.$

For the orthogonal group or the symplectic group, the projective homogeneous varieties have a similar description as varieties ofisotropicflags with respect to a given quadratic form or symplectic form. For any reductive groupGwith a Borel subgroupB,G/Bis called theflag varietyorflag manifoldofG.

Classification of split reductive groups

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.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 KillingandÉlie Cartanin 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 groupsGof type G₂and E₆had been constructed earlier, at least in the form of the abstract groupG(k), byL. E. Dickson.For example, the groupG₂is theautomorphism groupof anoctonion algebraoverk.By contrast, the Chevalley groups of type F₄,E₇,E₈over a field of positive characteristic were completely new.

More generally, the classification ofsplitreductive groups is the same over any field.A semisimple groupGover a fieldkis calledsimply connectedif every central isogeny from a semisimple group toGis an isomorphism. (ForGsemisimple over thecomplex numbers,being simply connected in this sense is equivalent toG(C) beingsimply connectedin the classical topology.) Chevalley's classification gives that, over any fieldk,there is a unique simply connected split semisimple groupGwith a given Dynkin diagram, with simple groups corresponding to the connected diagrams. At the other extreme, a semisimple group is ofadjoint typeif its center is trivial. The split semisimple groups overkwith given Dynkin diagram are exactly the groupsG/A,whereGis the simply connected group andAis ak-subgroup scheme of the center ofG.

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

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

$q(x_{1},\ldots,x_{2n+1})=x_{1}x_{2}+x_{3}x_{4}+\cdots +x_{2n-1}x_{2n}+x_{2n+1}^{2};$

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

$q(x_{1},\ldots,x_{2n})=x_{1}x_{2}+x_{3}x_{4}+\cdots +x_{2n-1}x_{2n}.$

Theouter automorphism groupof a split reductive groupGover a fieldkis isomorphic to the automorphism group of the root datum ofG.Moreover, the automorphism group ofGsplits as asemidirect product:

$\operatorname {Aut} (G)\cong \operatorname {Out} (G)\ltimes (G/Z)(k),$

whereZis the center ofG.For a split semisimple simply connected groupGover a field, the outer automorphism group ofGhas a simpler description: it is the automorphism group of the Dynkin diagram ofG.

Reductive group schemes

Agroup schemeGover a schemeSis calledreductiveif the morphismG→Sissmoothand affine, and every geometric fiber$G_{\overline {k}}$is reductive. (For a pointpinS,the corresponding geometric fiber means the base change ofGto an algebraic closure${\overline {k}}$of the residue field ofp.) Extending Chevalley's work,Michel Demazureand Grothendieck showed that split reductive group schemes over any nonempty schemeSare classified by root data.This statement includes the existence of Chevalley groups as group schemes overZ,and it says that every split reductive group over a schemeSis isomorphic to the base change of a Chevalley group fromZtoS.

Real reductive groups

In the context ofLie groupsrather than algebraic groups, areal reductive groupis a Lie groupGsuch that there is a linear algebraic groupLoverRwhose 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 forGconnected).

In particular, every connected semisimple Lie group (meaning that its Lie algebra is semisimple) is reductive. Also, the Lie groupRis 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 representationsand 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 groupGoverRmay 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 groupisomorphic 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 coverofSL(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/KofGby amaximal compact subgroupKis asymmetric spaceof non-compact type. In fact, every symmetric space of non-compact type arises this way. These are central examples inRiemannian geometryof 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 groupGover a fieldkthat is complete with respect to adiscrete valuation(such as thep-adic numbersQ_p), theaffine buildingXofGplays the role of the symmetric space. Namely,Xis asimplicial complexwith 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.

Representations of reductive groups

For a split reductive groupGover 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) ≅Zⁿwith 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.ThenBis the semidirect product ofTwith a smooth connected unipotent subgroupU.Define ahighest weight vectorin a representationVofGoverkto be a nonzero vectorvsuch thatBmaps the line spanned byvinto itself. ThenBacts on that line through its quotient groupT,by some element λ of the weight latticeX(T). Chevalley showed that every irreducible representation ofGhas 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.

There remains the problem of describing the irreducible representation with given highest weight. Forkof 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 bundleon the flag manifoldG/Bassociated to λ; this is a representation ofG.Forkof characteristic zero, theBorel–Weil theoremsays that the irreducible representationL(λ) is isomorphic to the Schur module ∇(λ). Furthermore, theWeyl character formulagives thecharacter(and in particular the dimension) of this representation.

For a split reductive groupGover a fieldkof positive characteristic, the situation is far more subtle, because representations ofGare 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.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 characteristicpofkis much bigger than theCoxeter numberofG,byHenning Andersen,Jens Jantzen,and Wolfgang Soergel (provingLusztig's conjecture in that case). Their character formula forplarge is based on theKazhdan–Lusztig polynomials,which are combinatorially complex.For any primep,Simon Riche andGeordie Williamsonconjectured the irreducible characters of a reductive group in terms of thep-Kazhdan-Lusztig polynomials, which are even more complex, but at least are computable.

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 theclassical groupsare:

• Every nondegenerate quadratic formqover a fieldkdetermines a reductive group G =SO(q). HereGis simple ifqhas dimensionnat least 3, since$G_{\overline {k}}$is isomorphic toSO(n) over an algebraic closure${\overline {k}}$.Thek-rank ofGis equal to theWitt indexofq(the maximum dimension of an isotropic subspace overk).So the simple groupGis split overkif and only ifqhas the maximum possible Witt index,$\lfloor n/2\rfloor$.
• Everycentral simple algebraAoverkdetermines a reductive groupG=SL(1,A), the kernel of thereduced normon thegroup of unitsA* (as an algebraic group overk). ThedegreeofAmeans the square root of the dimension ofAas ak-vector space. HereGis simple ifAhas degreenat least 2, since$G_{\overline {k}}$is isomorphic toSL(n) over${\overline {k}}$.IfAhas indexr(meaning thatAis isomorphic to the matrix algebraM_n/r(D) for adivision algebraDof degreeroverk), then thek-rank ofGis (n/r) − 1.So the simple groupGis split overkif and only ifAis a matrix algebra overk.

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

• Gis isotropic (that is,Gcontains a copy of the multiplicative groupG_moverk);
• Gcontains a parabolic subgroup overknot equal toG;
• Gcontains a copy of the additive groupG_aoverk.

Forkperfect, it is also equivalent to say thatG(k) contains aunipotentelement other than 1.

For a connected linear algebraic groupGover a local fieldkof characteristic zero (such as the real numbers), the groupG(k) iscompactin the classical topology (based on the topology ofk) if and only ifGis reductive and anisotropic.Example: the orthogonal groupSO(p,q)overRhas real rank min(p,q), and so it is anisotropic if and only ifporqis zero.

A reductive groupGover a fieldkis calledquasi-splitif it contains a Borel subgroup overk.A split reductive group is quasi-split. IfGis quasi-split overk,then any two Borel subgroups ofGare conjugate by some element ofG(k).Example: the orthogonal groupSO(p,q) overRis split if and only if |p−q| ≤ 1, and it is quasi-split if and only if |p−q| ≤ 2.

Structure of semisimple groups as abstract groups

For a simply connected split semisimple groupGover a fieldk,Robert Steinberggave an explicitpresentationof the abstract groupG(k).It is generated by copies of the additive group ofkindexed by the roots ofG(the root subgroups), with relations determined by the Dynkin diagram ofG.

For a simply connected split semisimple groupGover 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${\overline {k}}$-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).

For ak-simple algebraic groupG,Tits's simplicity theoremsays that the abstract groupG(k) is close to being simple, under mild assumptions. Namely, suppose thatGis isotropic overk,and suppose that the fieldkhas at least 4 elements. LetG(k)⁺be the subgroup of the abstract groupG(k) generated byk-points of copies of the additive groupG_aoverkcontained inG.(By the assumption thatGis isotropic overk,the groupG(k)⁺is nontrivial, and even Zariski dense inGifkis infinite.) Then the quotient group ofG(k)⁺by its center is simple (as an abstract group).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 whenGis split of typeA₁,B₂,orG₂,or non-split (that is, unitary) of typeA₂.Fork=F₃,the theorem holds except forGof typeA₁.

For ak-simple groupG,in order to understand the whole groupG(k), one can consider theWhitehead groupW(k,G)=G(k)/G(k)⁺.ForGsimply connected and quasi-split, the Whitehead group is trivial, and so the whole groupG(k) is simple modulo its center.More generally, theKneser–Tits problemasks 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, letDbe a division algebra with center ap-adic fieldk.Suppose that the dimension ofDoverkis 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.

Lattices and arithmetic groups

LetGbe a linear algebraic group over therational numbersQ.ThenGcan be extended to an affine group schemeGoverZ,and this determines an abstract groupG(Z). Anarithmetic groupmeans any subgroup ofG(Q) that iscommensurablewithG(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,alatticeinGmeans a discrete subgroup Γ ofGsuch 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 theoremsays, in particular: for a simple Lie groupGof real rank at least 2, every lattice inGis 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 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 theoremsays 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 theoremreduces 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 fieldkis determined up to isomorphism by its Tits index together with its anisotropic kernel, an associated anisotropic semisimplek-group.

For a reductive groupGover a fieldk,theabsolute Galois groupGal(k_s/k) acts (continuously) on the "absolute" Dynkin diagram ofG,that is, the Dynkin diagram ofGover aseparable closurek_s(which is also the Dynkin diagram ofGover an algebraic closure${\overline {k}}$). The Tits index ofGconsists 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 fieldkon a Dynkin diagram, there is a unique simply connected semisimple quasi-split groupHoverkwith the given action. (For a quasi-split group, every Galois orbit in the Dynkin diagram is circled.) Moreover, any other simply connected semisimple groupGoverkwith the given action is aninner formof the quasi-split groupH,meaning thatGis the group associated to an element of theGalois cohomologysetH¹(k,H/Z), whereZis the center ofH.In other words,Gis the twist ofHassociated to someH/Z-torsor overk,as discussed in the next section.

Example: Letqbe a nondegenerate quadratic form of even dimension 2nover a fieldkof characteristic not 2, withn≥ 5. (These restrictions can be avoided.) LetGbe the simple groupSO(q) overk.The absolute Dynkin diagram ofGis of type D_n,and so its automorphism group is of order 2, switching the two "legs" of the D_ndiagram. The action of the absolute Galois group ofkon the Dynkin diagram is trivial if and only if the signeddiscriminantdofqink*/(k*)²is trivial. Ifdis 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$\operatorname {Gal} (k_{s}/k({\sqrt {d}}))\subset \operatorname {Gal} (k_{s}/k)$.The groupGis split if and only ifqhas Witt indexn,the maximum possible, andGis quasi-split if and only ifqhas Witt index at leastn− 1.

Torsors and the Hasse principle

Atorsorfor an affine group schemeGover a fieldkmeans an affine schemeXoverkwith anactionofGsuch that$X_{\overline {k}}$is isomorphic to$G_{\overline {k}}$with the action of$G_{\overline {k}}$on itself by left translation. A torsor can also be viewed as aprincipal G-bundleoverkwith respect to thefppf topologyonk,or theétale topologyifGis smooth overk.Thepointed setof isomorphism classes ofG-torsors overkis calledH¹(k,G), in the language of Galois cohomology.

Torsors arise whenever one seeks to classifyformsof a given algebraic objectYover a fieldk,meaning objectsXoverkwhich become isomorphic toYover 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 dimensionnoverkare classified byH¹(k,O(n)), and central simple algebras of degreenoverkare 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 withabeliancoefficient groupsM,H^a(k,M). In this direction, Steinberg provedSerre's "Conjecture I": for a connected linear algebraic groupGover a perfect field ofcohomological dimensionat most 1,H¹(k,G) = 1.(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 IIpredicts that for a simply connected semisimple groupGover 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 Harderand Vladimir Chernousov (1989) proved theHasse principle:for a simply connected semisimple groupGoverk,the map

$H^{1}(k,G)\to \prod _{v}H^{1}(k_{v},G)$

is bijective.Herevruns over allplacesofk,andk_vis the corresponding local field (possiblyRorC). Moreover, the pointed setH¹(k_v,G) is trivial for every nonarchimidean local fieldk_v,and so only the real places ofkmatter. The analogous result for aglobal fieldkof positive characteristic was proved earlier by Harder (1975): for every simply connected semisimple groupGoverk,H¹(k,G) is trivial (sincekhas no real places).

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

$H^{1}(k,G)\to \prod _{v}H^{1}(k_{v},G)$

is injective.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.

See also

• Thegroups of Lie typeare 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

References

• 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,ISBN0-387-97370-2,MR1102012
• 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,MR0294349,S2CID119837998
• Chevalley, Claude(2005) [1958],Cartier, P.(ed.),Classification des groupes algébriques semi-simples,Collected Works, Vol. 3,Springer Nature,ISBN3-540-23031-9,MR2124841
• Conrad, Brian(2014),"Reductive group schemes"(PDF),Autour des schémas en groupes,vol. 1, Paris:Société Mathématique de France,pp. 93–444,ISBN978-2-85629-794-0,MR3309122
• Demazure, Michel;Gabriel, Pierre(1970),Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs,Paris: Masson,ISBN978-2225616662,MR0302656
• 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.ISBN978-2-85629-323-2.MR2867621.Revised and annotated edition of the 1970 original.
• Demazure, M.;Grothendieck, A.(1970).Schémas en groupes (SGA 3), II: 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.ISBN978-3540051800.MR0274459.
• 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.ISBN978-2-85629-324-9.MR2867622.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,ISBN978-285629-269-3,MR2605318
• Jantzen, Jens Carsten(2003) [1987],Representations of Algebraic Groups(2nd ed.),American Mathematical Society,ISBN978-0-8218-3527-2,MR2015057
• Milne, J. S.(2017),Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field,Cambridge University Press,doi:10.1017/9781316711736,ISBN978-1107167483,MR3729270
• Platonov, Vladimir;Rapinchuk, Andrei (1994),Algebraic Groups and Number Theory,Academic Press,ISBN0-12-558180-7,MR1278263
• 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,ISBN978-2-85629-880-0
• Springer, Tonny A.(1979),"Reductive groups",Automorphic Forms, Representations, andL-functions,vol. 1,American Mathematical Society,pp. 3–27,ISBN0-8218-3347-2,MR0546587
• 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,ISBN978-0-8176-4021-7,MR1642713
• Steinberg, Robert(1965),"Regular elements of semisimple algebraic groups",Publications Mathématiques de l'IHÉS,25:49–80,doi:10.1007/bf02684397,MR0180554,S2CID55638217
• Steinberg, Robert(2016) [1968],Lectures on Chevalley Groups,University Lecture Series, vol. 66,American Mathematical Society,doi:10.1090/ulect/066,ISBN978-1-4704-3105-1,MR3616493
• Tits, Jacques(1964), "Algebraic and abstract simple groups",Annals of Mathematics,80(2): 313–329,doi:10.2307/1970394,JSTOR1970394,MR0164968

External links

• 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 générauxRevised and annotated edition of the 1970 original.