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v t e

Inmathematics, alinear algebraic group is asubgroup of thegroup ofinvertible${\displaystyle n\times n}$matrices (undermatrix multiplication) that is defined bypolynomial equations. An example is theorthogonal group, defined by the relation${\displaystyle M^{T}M=I_n}$ where${\displaystyle M^T}$ is thetranspose of${\displaystyle M}$.

ManyLie groups can be viewed as linear algebraic groups over thefield ofreal orcomplex numbers. (For example, everycompact Lie group can be regarded as a linear algebraic group overR (necessarilyR-anisotropic and reductive), as can many noncompact groups such as thesimple Lie groupSL(n,R).) The simple Lie groups were classified byWilhelm Killing andÉlie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups includeMaurer,Chevalley, andKolchin (1948). In the 1950s,Armand Borel constructed much of the theory of algebraic groups as it exists today.

One of the first uses for the theory was to define theChevalley groups.

For apositive integer${\displaystyle n}$, thegeneral linear group${\displaystyle GL(n)}$ over a field${\displaystyle k}$, consisting of all invertible${\displaystyle n\times n}$ matrices, is a linear algebraic group over${\displaystyle k}$. It contains the subgroups

${\displaystyle U\subset B\subset GL(n)}$

consisting of matrices of the form, resp.,

and.

The group${\displaystyle U}$ is an example of aunipotent linear algebraic group, the group${\displaystyle B}$ is an example of asolvable algebraic group called theBorel subgroup of${\displaystyle GL(n)}$. It is a consequence of theLie-Kolchin theorem that any connected solvable subgroup of${\displaystyle \mathrm{G}\mathrm{L}(n)}$ is conjugated into${\displaystyle B}$. Any unipotent subgroup can be conjugated into${\displaystyle U}$.

Another algebraic subgroup of${\displaystyle \mathrm{G}\mathrm{L}(n)}$ is thespecial linear group${\displaystyle \mathrm{S}\mathrm{L}(n)}$ of matrices with determinant 1.

The group${\displaystyle \mathrm{G}\mathrm{L}(1)}$ is called themultiplicative group, usually denoted by${\displaystyle {\mathbf{G}}_{\mathrm{m}}}$. The group of${\displaystyle k}$-points${\displaystyle {\mathbf{G}}_{\mathrm{m}}(k)}$ is the multiplicative group${\displaystyle k^{\ast }}$ of nonzero elements of the field${\displaystyle k}$. Theadditive group${\displaystyle {\mathbf{G}}_{\mathrm{a}}}$, whose${\displaystyle k}$-points are isomorphic to the additive group of${\displaystyle k}$, can also be expressed as a matrix group, for example as the subgroup${\displaystyle U}$ in${\displaystyle \mathrm{G}\mathrm{L}(2)}$ :

These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of theirlinear representations (as algebraic groups). Every representation of the multiplicative group${\displaystyle {\mathbf{G}}_{\mathrm{m}}}$ is adirect sum ofirreducible representations. (Its irreducible representations all have dimension 1, of the form${\displaystyle x\mapsto x^n}$ for an integer${\displaystyle n}$.) By contrast, the only irreducible representation of the additive group${\displaystyle {\mathbf{G}}_{\mathrm{a}}}$ is the trivial representation. So every representation of${\displaystyle {\mathbf{G}}_{\mathrm{a}}}$ (such as the 2-dimensional representation above) is an iteratedextension of trivial representations, not a direct sum (unless the representation is trivial). The structure theory of linear algebraic groups analyzes any linear algebraic group in terms of these two basic groups and their generalizations, tori and unipotent groups, as discussed below.

For analgebraically closed fieldk, much of the structure of analgebraic varietyX overk is encoded in its setX(k) ofk-rational points, which allows an elementary definition of a linear algebraic group. First, define a function from the abstract groupGL(n,k) tok to beregular if it can be written as a polynomial in the entries of ann×n matrixA and in 1/det(A), where det is thedeterminant. Then alinear algebraic groupG over an algebraically closed fieldk is a subgroupG(k) of the abstract groupGL(n,k) for some natural numbern such thatG(k) is defined by the vanishing of some set of regular functions.

For an arbitrary fieldk, algebraic varieties overk are defined as a special case ofschemes overk. In that language, alinear algebraic groupG over a fieldk is asmooth closed subgroup scheme ofGL(n) overk for some natural numbern. In particular,G is defined by the vanishing of some set ofregular functions onGL(n) overk, and these functions must have the property that for every commutativek-algebraR,G(R) is a subgroup of the abstract groupGL(n,R). (Thus an algebraic groupG overk is not just the abstract groupG(k), but rather the whole family of groupsG(R) for commutativek-algebrasR; this is the philosophy of describing a scheme by itsfunctor of points.)

In either language, one has the notion of ahomomorphism of linear algebraic groups. For example, whenk is algebraically closed, a homomorphism fromG ⊂GL(m) toH ⊂GL(n) is a homomorphism of abstract groupsG(k) →H(k) which is defined by regular functions onG. This makes the linear algebraic groups overk into acategory. In particular, this defines what it means for two linear algebraic groups to beisomorphic.

In the language of schemes, a linear algebraic groupG over a fieldk is in particular agroup scheme overk, meaning a scheme overk together with ak-point 1 ∈G(k) and morphisms

${\displaystyle m:G{\times }_{k}G\to G,i:G\to G}$

overk which satisfy the usual axioms for the multiplication and inverse maps in a group (associativity, identity, inverses). A linear algebraic group is also smooth and offinite type overk, and it isaffine (as a scheme). Conversely, every affine group schemeG of finite type over a fieldk has afaithful representation intoGL(n) overk for somen.^[1] An example is the embedding of the additive groupG_a intoGL(2), as mentioned above. As a result, one can think of linear algebraic groups either as matrix groups or, more abstractly, as smooth affine group schemes over a field. (Some authors use "linear algebraic group" to mean any affine group scheme of finite type over a field.)

For a full understanding of linear algebraic groups, one has to consider more general (non-smooth) group schemes. For example, letk be an algebraically closed field ofcharacteristicp > 0. Then the homomorphismf:G_m →G_m defined byx ↦x^p induces an isomorphism of abstract groupsk* →k*, butf is not an isomorphism of algebraic groups (becausex^1/p is not a regular function). In the language of group schemes, there is a clearer reason whyf is not an isomorphism:f is surjective, but it has nontrivialkernel, namely thegroup scheme μ_p ofpth roots of unity. This issue does not arise in characteristic zero. Indeed, every group scheme of finite type over a fieldk of characteristic zero is smooth overk.^[2] A group scheme of finite type over any fieldk is smooth overk if and only if it isgeometrically reduced, meaning that thebase change${\displaystyle G_{\overline{k}}}$ isreduced, where${\displaystyle \overline{k}}$ is analgebraic closure ofk.^[3]

Since an affine schemeX is determined by itsringO(X) of regular functions, an affine group schemeG over a fieldk is determined by the ringO(G) with its structure of aHopf algebra (coming from the multiplication and inverse maps onG). This gives anequivalence of categories (reversing arrows) between affine group schemes overk and commutative Hopf algebras overk. For example, the Hopf algebra corresponding to the multiplicative groupG_m =GL(1) is theLaurent polynomial ringk[x,x⁻¹], with comultiplication given by

${\displaystyle x\mapsto x\otimes x.}$

For a linear algebraic groupG over a fieldk, theidentity componentG^o (theconnected component containing the point 1) is anormal subgroup of finiteindex. So there is agroup extension

${\displaystyle 1\to G^{\circ }\to G\to F\to 1,}$

whereF is a finite algebraic group. (Fork algebraically closed,F can be identified with an abstract finite group.) Because of this, the study of algebraic groups mostly focuses on connected groups.

Various notions fromabstract group theory can be extended to linear algebraic groups. It is straightforward to define what it means for a linear algebraic group to becommutative,nilpotent, orsolvable, by analogy with the definitions in abstract group theory. For example, a linear algebraic group issolvable if it has acomposition series of linear algebraic subgroups such that the quotient groups are commutative. Also, thenormalizer, thecenter, and thecentralizer of a closed subgroupH of a linear algebraic groupG are naturally viewed as closed subgroup schemes ofG. If they are smooth overk, then they are linear algebraic groups as defined above.

One may ask to what extent the properties of a connected linear algebraic groupG over a fieldk are determined by the abstract groupG(k). A useful result in this direction is that if the fieldk isperfect (for example, of characteristic zero),or ifG is reductive (as defined below), thenG isunirational overk. Therefore, if in additionk is infinite, the groupG(k) isZariski dense inG.^[4] For example, under the assumptions mentioned,G is commutative, nilpotent, or solvable if and only ifG(k) has the corresponding property.

The assumption of connectedness cannot be omitted in these results. For example, letG be the group μ₃ ⊂GL(1) of cube roots of unity over therational numbersQ. ThenG is a linear algebraic group overQ for whichG(Q) = 1 is not Zariski dense inG, because${\displaystyle G(\overline{\mathbf{Q}})}$ is a group of order 3.

Over an algebraically closed field, there is a stronger result about algebraic groups as algebraic varieties: every connected linear algebraic group over an algebraically closed field is arational variety.^[5]

TheLie algebra${\displaystyle \mathfrak{g}}$ of an algebraic groupG can be defined in several equivalent ways: as thetangent spaceT₁(G) at the identity element 1 ∈G(k), or as the space of left-invariantderivations. Ifk is algebraically closed, a derivationD:O(G) →O(G) overk of the coordinate ring ofG isleft-invariant if

${\displaystyle D{\lambda }_x={\lambda }_{x}D}$

for everyx inG(k), where λ_x:O(G) →O(G) is induced by left multiplication byx. For an arbitrary fieldk, left invariance of a derivation is defined as an analogous equality of two linear mapsO(G) →O(G) ⊗O(G).^[6] The Lie bracket of two derivations is defined by [D₁,D₂] =D₁D₂ −D₂D₁.

The passage fromG to${\displaystyle \mathfrak{g}}$ is thus a process ofdifferentiation. For an elementx ∈G(k), the derivative at 1 ∈G(k) of theconjugation mapG →G,g ↦xgx⁻¹, is anautomorphism of${\displaystyle \mathfrak{g}}$, giving theadjoint representation:

${\displaystyle \mathrm{Ad}:G\to \mathrm{Aut}(\mathfrak{g}).}$

Over a field of characteristic zero, a connected subgroupH of a linear algebraic groupG is uniquely determined by its Lie algebra${\displaystyle \mathfrak{h}\subset \mathfrak{g}}$.^[7] But not every Lie subalgebra of${\displaystyle \mathfrak{g}}$ corresponds to an algebraic subgroup ofG, as one sees in the example of the torusG = (G_m)² overC. In positive characteristic, there can be many different connected subgroups of a groupG with the same Lie algebra (again, the torusG = (G_m)² provides examples). For these reasons, although the Lie algebra of an algebraic group is important, the structure theory of algebraic groups requires more global tools.

For an algebraically closed fieldk, a matrixg inGL(n,k) is calledsemisimple if it isdiagonalizable, andunipotent if the matrixg − 1 isnilpotent. Equivalently,g is unipotent if alleigenvalues ofg are equal to 1. TheJordan canonical form for matrices implies that every elementg ofGL(n,k) can be written uniquely as a productg =g_ssg_u such thatg_ss is semisimple,g_u is unipotent, andg_ss andg_ucommute with each other.

For any fieldk, an elementg ofGL(n,k) is said to be semisimple if it becomes diagonalizable over the algebraic closure ofk. If the fieldk is perfect, then the semisimple and unipotent parts ofg also lie inGL(n,k). Finally, for any linear algebraic groupG ⊂GL(n) over a fieldk, define ak-point ofG to be semisimple or unipotent if it is semisimple or unipotent inGL(n,k). (These properties are in fact independent of the choice of a faithful representation ofG.) If the fieldk is perfect, then the semisimple and unipotent parts of ak-point ofG are automatically inG. That is (theJordan decomposition): every elementg ofG(k) can be written uniquely as a productg =g_ssg_u inG(k) such thatg_ss is semisimple,g_u is unipotent, andg_ss andg_u commute with each other.^[8] This reduces the problem of describing theconjugacy classes inG(k) to the semisimple and unipotent cases.

Atorus over an algebraically closed fieldk means a group isomorphic to (G_m)ⁿ, theproduct ofn copies of the multiplicative group overk, for some natural numbern. For a linear algebraic groupG, amaximal torus inG means a torus inG that is not contained in any bigger torus. For example, the group of diagonal matrices inGL(n) overk is a maximal torus inGL(n), isomorphic to (G_m)ⁿ. A basic result of the theory is that any two maximal tori in a groupG over an algebraically closed fieldk areconjugate by some element ofG(k).^[9] Therank ofG means the dimension of any maximal torus.

For an arbitrary fieldk, atorusT overk means a linear algebraic group overk whose base change${\displaystyle T_{\overline{k}}}$ to the algebraic closure ofk is isomorphic to (G_m)ⁿ over${\displaystyle \overline{k}}$, for some natural numbern. Asplit torus overk means a group isomorphic to (G_m)ⁿ overk for somen. An example of a non-split torus over the real numbersR is

${\displaystyle T=\{(x,y)\in A_{\mathbf{R}}^2:x^2+y^2=1\},}$

with group structure given by the formula for multiplying complex numbersx+iy. HereT is a torus of dimension 1 overR. It is not split, because its group of real pointsT(R) is thecircle group, which is not isomorphic even as an abstract group toG_m(R) =R*.

Every point of a torus over a fieldk is semisimple. Conversely, ifG is a connected linear algebraic group such that every element of${\displaystyle G(\overline{k})}$ is semisimple, thenG is a torus.^[10]

For a linear algebraic groupG over a general fieldk, one cannot expect all maximal tori inG overk to be conjugate by elements ofG(k). For example, both the multiplicative groupG_m and the circle groupT above occur as maximal tori inSL(2) overR. However, it is always true that any twomaximal split tori inG overk (meaning split tori inG that are not contained in a biggersplit torus) are conjugate by some element ofG(k).^[11] As a result, it makes sense to define thek-rank orsplit rank of a groupG overk as the dimension of any maximal split torus inG overk.

For any maximal torusT in a linear algebraic groupG over a fieldk, Grothendieck showed that${\displaystyle T_{\overline{k}}}$ is a maximal torus in${\displaystyle G_{\overline{k}}}$.^[12] It follows that any two maximal tori inG over a fieldk have the same dimension, although they need not be isomorphic.

LetU_n be the group of upper-triangular matrices inGL(n) with diagonal entries equal to 1, over a fieldk. A group scheme over a fieldk (for example, a linear algebraic group) is calledunipotent if it is isomorphic to a closed subgroup scheme ofU_n for somen. It is straightforward to check that the groupU_n is nilpotent. As a result, every unipotent group scheme is nilpotent.

A linear algebraic groupG over a fieldk is unipotent if and only if every element of${\displaystyle G(\overline{k})}$ is unipotent.^[13]

The groupB_n of upper-triangular matrices inGL(n) is asemidirect product

${\displaystyle B_n=T_n⋉U_n,}$

whereT_n is the diagonal torus (G_m)ⁿ. More generally, every connected solvable linear algebraic group is a semidirect product of a torus with a unipotent group,T ⋉U.^[14]

A smooth connected unipotent group over a perfect fieldk (for example, an algebraically closed field) has a composition series with all quotient groups isomorphic to the additive groupG_a.^[15]

TheBorel subgroups are important for the structure theory of linear algebraic groups. For a linear algebraic groupG over an algebraically closed fieldk, a Borel subgroup ofG means a maximal smooth connected solvable subgroup. For example, one Borel subgroup ofGL(n) is the subgroupB ofupper-triangular matrices (all entries below the diagonal are zero).

A basic result of the theory is that any two Borel subgroups of a connected groupG over an algebraically closed fieldk are conjugate by some element ofG(k).^[16] (A standard proof uses theBorel fixed-point theorem: for a connected solvable groupG acting on aproper varietyX over an algebraically closed fieldk, there is ak-point inX which is fixed by the action ofG.) The conjugacy of Borel subgroups inGL(n) amounts to theLie–Kolchin theorem: every smooth connected solvable subgroup ofGL(n) is conjugate to a subgroup of the upper-triangular subgroup inGL(n).

For an arbitrary fieldk, a Borel subgroupB ofG is defined to be a subgroup overk such that, over an algebraic closure${\displaystyle \overline{k}}$ ofk,${\displaystyle B_{\overline{k}}}$ is a Borel subgroup of${\displaystyle G_{\overline{k}}}$. ThusG may or may not have a Borel subgroup overk.

For a closed subgroup schemeH ofG, thequotient spaceG/H is a smoothquasi-projective scheme overk.^[17] A smooth subgroupP of a connected groupG is calledparabolic ifG/P isprojective overk (or equivalently, proper overk). An important property of Borel subgroupsB is thatG/B is a projective variety, called theflag variety ofG. That is, Borel subgroups are parabolic subgroups. More precisely, fork algebraically closed, the Borel subgroups are exactly the minimal parabolic subgroups ofG; conversely, every subgroup containing a Borel subgroup is parabolic.^[18] So one can list all parabolic subgroups ofG (up to conjugation byG(k)) by listing all the linear algebraic subgroups ofG that contain a fixed Borel subgroup. For example, the subgroupsP ⊂GL(3) overk that contain the Borel subgroupB of upper-triangular matrices areB itself, the whole groupGL(3), and the intermediate subgroups

and

The correspondingprojective homogeneous varietiesGL(3)/P are (respectively): theflag manifold of all chains of linear subspaces

${\displaystyle 0\subset V_1\subset V_2\subset A_k^3}$

withV_i of dimensioni; a point; theprojective spaceP² of lines (1-dimensionallinear subspaces) inA³; and the dual projective spaceP² of planes inA³.

A connected linear algebraic groupG over an algebraically closed field is calledsemisimple if every smooth connected solvable normal subgroup ofG is trivial. More generally, a connected linear algebraic groupG over an algebraically closed field is calledreductive if every smooth connected unipotent normal subgroup ofG is trivial.^[19] (Some authors do not require reductive groups to be connected.) A semisimple group is reductive. A groupG over an arbitrary fieldk is called semisimple or reductive if${\displaystyle G_{\overline{k}}}$ is semisimple or reductive. For example, the groupSL(n) ofn ×n matrices with determinant 1 over any fieldk is semisimple, whereas a nontrivial torus is reductive but not semisimple. Likewise,GL(n) is reductive but not semisimple (because its centerG_m is a nontrivial smooth connected solvable normal subgroup).

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.^[20]

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.^[21] (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integern at least 2 and any fieldk, the groupSL(n) overk is simple, and its center is the group scheme μ_n ofnth roots of unity.

Every connected linear algebraic groupG over a perfect fieldk is (in a unique way) an extension of a reductive groupR by a smooth connected unipotent groupU, called theunipotent radical ofG:

${\displaystyle 1\to U\to G\to R\to 1.}$

Ifk has characteristic zero, then one has the more preciseLevi decomposition: every connected linear algebraic groupG overk is a semidirect product${\displaystyle R⋉U}$ of a reductive group by a unipotent group.^[22]

Reductive groups include the most important linear algebraic groups in practice, such as theclassical groups:GL(n),SL(n), theorthogonal groupsSO(n) and thesymplectic groupsSp(2n). On the other hand, the definition of reductive groups is quite "negative", and it is not clear that one can expect to say much about them. Remarkably,Claude Chevalley gave a complete classification of the reductive groups over an algebraically closed field: they are determined byroot data.^[23] In particular, simple groups over an algebraically closed fieldk are classified (up to quotients by finite central subgroup schemes) by theirDynkin diagrams. It is striking that this classification is independent of the characteristic ofk. For example, theexceptional Lie groupsG₂,F₄,E₆,E₇, andE₈ can be defined in any characteristic (and even as group schemes overZ). Theclassification of finite simple groups says that most finite simple groups arise as the group ofk-points of a simple algebraic group over a finite fieldk, or as minor variants of that construction.

Every reductive group over a field is the quotient by a finite central subgroup scheme of the product of a torus and some simple groups. For example,

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

For an arbitrary fieldk, a reductive groupG is calledsplit if it contains a split maximal torus overk (that is, a split torus inG which remains maximal over an algebraic closure ofk). For example,GL(n) is a split reductive group over any fieldk. Chevalley showed that the classification ofsplit reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. For example, every nondegeneratequadratic formq over a fieldk determines a reductive groupSO(q), and everycentral simple algebraA overk determines a reductive groupSL₁(A). 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 asnumber fields, but for arbitrary fields there are many open questions.

One reason for the importance of reductive groups comes from representation theory. Every irreducible representation of a unipotent group is trivial. More generally, for any linear algebraic groupG written as an extension

${\displaystyle 1\to U\to G\to R\to 1}$

withU unipotent andR reductive, every irreducible representation ofG factors throughR.^[24] This focuses attention on the representation theory of reductive groups. (To be clear, the representations considered here are representations ofGas an algebraic group. Thus, for a groupG over a fieldk, the representations are onk-vector spaces, and the action ofG is given by regular functions. It is an important but different problem to classifycontinuous representations of the groupG(R) for a real reductive groupG, or similar problems over other fields.)

Chevalley showed that the irreducible representations of a split reductive group over a fieldk are finite-dimensional, and they are indexed bydominant weights.^[25] This is the same as what happens in the representation theory of compact connected Lie groups, or the finite-dimensional representation theory of complexsemisimple Lie algebras. Fork of characteristic zero, all these theories are essentially equivalent. In particular, every representation of a reductive groupG over a field of characteristic zero is a direct sum of irreducible representations, and ifG is split, thecharacters of the irreducible representations are given by theWeyl character formula. TheBorel–Weil theorem gives a geometric construction of the irreducible representations of a reductive groupG in characteristic zero, as spaces of sections ofline bundles over the flag manifoldG/B.

The representation theory of reductive groups (other than tori) over a field of positive characteristicp is less well understood. In this situation, a representation need not be a direct sum of irreducible representations. And although irreducible representations are indexed by dominant weights, the dimensions and characters of the irreducible representations are known only in some cases. Andersen, Jantzen and Soergel (1994) determined these characters (provingLusztig's conjecture) when the characteristicp is sufficiently large compared to theCoxeter number of the group. For small primesp, there is not even a precise conjecture.

Anaction of a linear algebraic groupG on a variety (or scheme)X over a fieldk is a morphism

${\displaystyle G{\times }_{k}X\to X}$

that satisfies the axioms of agroup action. As in other types of group theory, it is important to study group actions, since groups arise naturally as symmetries of geometric objects.

Part of the theory of group actions isgeometric invariant theory, which aims to construct a quotient varietyX/G, describing the set oforbits of a linear algebraic groupG onX as an algebraic variety. Various complications arise. For example, ifX is an affine variety, then one can try to constructX/G asSpec of thering of invariantsO(X)^G. However,Masayoshi Nagata showed that the ring of invariants need not be finitely generated as ak-algebra (and so Spec of the ring is a scheme but not a variety), a negative answer toHilbert's 14th problem. In the positive direction, the ring of invariants is finitely generated ifG is reductive, byHaboush's theorem, proved in characteristic zero byHilbert and Nagata.

Geometric invariant theory involves further subtleties when a reductive groupG acts on a projective varietyX. In particular, the theory defines open subsets of "stable" and "semistable" points inX, with the quotient morphism only defined on the set of semistable points.

Linear algebraic groups admit variants in several directions. Dropping the existence of the inverse map${\displaystyle i:G\to G}$, one obtains the notion of a linear algebraicmonoid.^[26]

For a linear algebraic groupG over the real numbersR, the group of real pointsG(R) is aLie group, essentially because real polynomials, which describe the multiplication onG, aresmooth functions. Likewise, for a linear algebraic groupG overC,G(C) is acomplex Lie group. Much of the theory of algebraic groups was developed by analogy with Lie groups.

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

• A Lie group with an infinite group of components G/G^o cannot be realized as a linear algebraic group.
• An algebraic groupG overR may be connected as an algebraic group while the Lie groupG(R) is not connected, and likewise forsimply connected groups. For example, the algebraic groupSL(2) is simply connected over any field, whereas the Lie groupSL(2,R) hasfundamental group isomorphic to the integersZ. The double coverH ofSL(2,R), known as themetaplectic group, is a Lie group that cannot be viewed as a linear algebraic group overR. More strongly,H has no faithful finite-dimensional representation.
• Anatoly Maltsev showed that every simply connected nilpotent Lie group can be viewed as a unipotent algebraic groupG overR in a unique way.^[27] (As a variety,G is isomorphic toaffine space of some dimension overR.) By contrast, there are simply connected solvable Lie groups that cannot be viewed as real algebraic groups. For example, theuniversal coverH of the semidirect productS¹ ⋉R² has center isomorphic toZ, which is not a linear algebraic group, and soH cannot be viewed as a linear algebraic group overR.

Algebraic groups which are not affine behave very differently. In particular, a smooth connected group scheme which is a projective variety over a field is called anabelian variety. In contrast to linear algebraic groups, every abelian variety is commutative. Nonetheless, abelian varieties have a rich theory. Even the case ofelliptic curves (abelian varieties of dimension 1) is central tonumber theory, with applications including the proof ofFermat's Last Theorem.

The finite-dimensional representations of an algebraic groupG, together with thetensor product of representations, form atannakian category Rep_G. In fact, tannakian categories with a "fiber functor" over a field are equivalent to affine group schemes. (Every affine group scheme over a fieldk ispro-algebraic in the sense that it is aninverse limit of affine group schemes of finite type overk.^[28]) For example, theMumford–Tate group and themotivic Galois group are constructed using this formalism. Certain properties of a (pro-)algebraic groupG can be read from its category of representations. For example, over a field of characteristic zero, Rep_G is asemisimple category if and only if the identity component ofG is pro-reductive.^[29]

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

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• De Medts, Tom (2019),Linear Algebraic Groups (course notes)(PDF), Ghent University
• Humphreys, James E. (1975),Linear Algebraic Groups, Springer,ISBN 0-387-90108-6,MR 0396773
• Kolchin, E. R. (1948), "Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations",Annals of Mathematics, Second Series,49 (1):1–42,doi:10.2307/1969111,ISSN 0003-486X,JSTOR 1969111,MR 0024884
• Milne, J. S. (2017),Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field,Cambridge University Press,ISBN 978-1107167483,MR 3729270
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• "Linear algebraic group",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Linear algebraic groups

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