# Algebraic group

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Algebraic variety with a group structure

Not to be confused with the [variety of groups](/source/Variety_(universal_algebra)).

Algebraic structure → Group theory Group theory Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic Discrete groups Lattices Integers ( Z {\displaystyle \mathbb {Z} } ) Free group Modular groups PSL(2, Z {\displaystyle \mathbb {Z} } ) SL(2, Z {\displaystyle \mathbb {Z} } ) Arithmetic group Lattice Hyperbolic group Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve v t e

In [mathematics](/source/Mathematics), an **algebraic group** is an [algebraic variety](/source/Algebraic_variety) endowed with a [group](/source/Group_(mathematics)) structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to [algebraic geometry](/source/Algebraic_geometry) and [group theory](/source/Group_theory).

Many groups of [geometric transformations](/source/Geometric_transformation) are algebraic groups, including [orthogonal groups](/source/Orthogonal_group), [general linear groups](/source/General_linear_group), [projective groups](/source/Projective_linear_group), [Euclidean groups](/source/Euclidean_group), etc. Many [matrix groups](/source/Matrix_group) are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as [elliptic curves](/source/Elliptic_curve) and [Jacobian varieties](/source/Jacobian_varieties).

An important class of algebraic groups is given by the [affine algebraic groups](/source/Affine_algebraic_group), those whose underlying algebraic variety is an [affine variety](/source/Affine_variety); they are exactly the algebraic subgroups of the [general linear group](/source/General_linear_group), and are therefore also called *linear algebraic groups*.[1] Another class is formed by the [abelian varieties](/source/Abelian_varieties), which are the algebraic groups whose underlying variety is a [projective variety](/source/Projective_variety). [Chevalley's structure theorem](/source/Chevalley's_structure_theorem) states that every algebraic group can be constructed from groups in those two families.

## Definitions

Formally, an algebraic group over a field k {\displaystyle k} is an algebraic variety G {\displaystyle \mathrm {G} } over k {\displaystyle k} , together with a distinguished element e ∈ G ( k ) {\displaystyle e\in \mathrm {G} (k)} (the [neutral element](/source/Neutral_element)), and [regular maps](/source/Regular_map_(algebraic_geometry)) G × G → G {\displaystyle \mathrm {G} \times \mathrm {G} \to \mathrm {G} } (the multiplication operation) and G → G {\displaystyle \mathrm {G} \to \mathrm {G} } (the inversion operation) that satisfy the group axioms.[2]

### Examples

- The *additive group*: the [affine line](/source/Affine_line) A 1 {\displaystyle \mathbb {A} ^{1}} endowed with addition and opposite as group operations is an algebraic group. It is called the additive group (because its k {\displaystyle k} -points are isomorphic as a group to the additive group of k {\displaystyle k} ), and usually denoted by G a {\displaystyle \mathrm {G} _{a}} .

- The *multiplicative group*: Let G m {\displaystyle \mathrm {G} _{m}} be the affine variety defined by the equation x y = 1 {\displaystyle xy=1} in the affine plane A 2 {\displaystyle \mathbb {A} ^{2}} . The functions ( ( x , y ) , ( x ′ , y ′ ) ) ↦ ( x x ′ , y y ′ ) {\displaystyle ((x,y),(x',y'))\mapsto (xx',yy')} and ( x , y ) ↦ ( x − 1 , y − 1 ) {\displaystyle (x,y)\mapsto (x^{-1},y^{-1})} are regular on G m {\displaystyle \mathrm {G} _{m}} , and they satisfy the group axioms (with neutral element ( 1 , 1 ) {\displaystyle (1,1)} ). The algebraic group G m {\displaystyle \mathrm {G} _{m}} is called the multiplicative group, because its k {\displaystyle k} -points are isomorphic to the multiplicative group of the field k {\displaystyle k} (an isomorphism is given by x ↦ ( x , x − 1 ) {\displaystyle x\mapsto (x,x^{-1})} ; note that the subset of invertible elements does not define an algebraic subvariety in A 1 {\displaystyle \mathbb {A} ^{1}} ).

- The [special linear group](/source/Special_linear_group) S L n {\displaystyle \mathrm {SL} _{n}} is an algebraic group: it is given by the algebraic equation det ( g ) = 1 {\displaystyle \det(g)=1} in the affine space A n 2 {\displaystyle \mathbb {A} ^{n^{2}}} (identified with the space of n {\displaystyle n} -by- n {\displaystyle n} matrices), multiplication of matrices is regular and the formula for the inverse in terms of the [adjugate matrix](/source/Adjugate_matrix) shows that inversion is regular as well on matrices with determinant 1.

- The [general linear group](/source/General_linear_group) G L n {\displaystyle \mathrm {GL} _{n}} of [invertible matrices](/source/Invertible_matrices) over a field k {\displaystyle k} is an algebraic group. It can be realized as a subvariety in A n 2 + 1 {\displaystyle \mathbb {A} ^{n^{2}+1}} in much the same way as the multiplicative group in the previous example.[3]

- A non-singular [cubic curve](/source/Cubic_curve) in the projective plane P 2 {\displaystyle \mathbb {P} ^{2}} with a specified point can be endowed with a geometrically defined group law that makes it into an algebraic group (see [elliptic curve](/source/Elliptic_curve)).

### Related definitions

An **algebraic subgroup** of an algebraic group G {\displaystyle \mathrm {G} } is a [subvariety](/source/Algebraic_variety#Subvariety) H {\displaystyle \mathrm {H} } of G {\displaystyle \mathrm {G} } that is also a subgroup of G {\displaystyle \mathrm {G} } (that is, the maps G × G → G {\displaystyle \mathrm {G} \times \mathrm {G} \to \mathrm {G} } and G → G {\displaystyle \mathrm {G} \to \mathrm {G} } defining the group structure map H × H {\displaystyle \mathrm {H} \times \mathrm {H} } and H {\displaystyle \mathrm {H} } , respectively, into H {\displaystyle \mathrm {H} } ).

A *morphism* between two algebraic groups G , G ′ {\displaystyle \mathrm {G} ,\mathrm {G} '} is a regular map G → G ′ {\displaystyle \mathrm {G} \to \mathrm {G} '} that is also a group homomorphism. Its kernel is an algebraic subgroup of G {\displaystyle \mathrm {G} } , and its image is an algebraic subgroup of G ′ {\displaystyle \mathrm {G} '} .[4]

[Quotients](/source/Quotient_group) in the category of algebraic groups are more delicate to deal with. An algebraic subgroup is said to be *normal* if it is stable under every [inner automorphism](/source/Inner_automorphism) (which are regular maps). If H {\displaystyle \mathrm {H} } is a normal algebraic subgroup of G {\displaystyle \mathrm {G} } , then there exists an algebraic group G / H {\displaystyle \mathrm {G} /\mathrm {H} } and a surjective morphism π : G → G / H {\displaystyle \pi :\mathrm {G} \to \mathrm {G} /\mathrm {H} } such that H {\displaystyle \mathrm {H} } is the kernel of π {\displaystyle \pi } .[5] Note that if the field k {\displaystyle k} is not algebraically closed, then the morphism of groups G ( k ) → G ( k ) / H ( k ) {\displaystyle \mathrm {G} (k)\to \mathrm {G} (k)/\mathrm {H} (k)} may not be surjective (the defect of surjectivity is measured by [Galois cohomology](/source/Galois_cohomology)).

### Lie algebra of an algebraic group

Similarly to the [Lie group–Lie algebra correspondence](/source/Lie_group%E2%80%93Lie_algebra_correspondence), to an algebraic group over a field k {\displaystyle k} is associated a [Lie algebra](/source/Lie_algebra) over k {\displaystyle k} . In positive [characteristic](/source/Characteristic_(algebra)), this is a [restricted Lie algebra](/source/Restricted_Lie_algebra). As a vector space, the Lie algebra is isomorphic to the [tangent space](/source/Tangent_space) at the identity element. The Lie bracket can be constructed from its interpretation as a space of [derivations](/source/Derivation_(differential_algebra)).[6]

### Alternative definitions

A more sophisticated and more general definition is that of a [group scheme](/source/Group_scheme) (which can more generally be defined over [commutative rings](/source/Commutative_ring)). An algebraic group over a field k {\displaystyle k} is then a group scheme over k {\displaystyle k} that is simultaneously a k {\displaystyle k} -variety.

Yet another definition of the concept is to say that an algebraic group over k {\displaystyle k} is a [group object](/source/Group_object) in the [category](/source/Category_(mathematics)) of algebraic varieties over k {\displaystyle k} .

## Affine algebraic groups

Main article: [Linear algebraic group](/source/Linear_algebraic_group)

An algebraic group is said to be *affine* if its underlying algebraic variety is an affine variety. Among the examples above, the additive, multiplicative, general linear, and special linear groups are affine. Using the action of an affine algebraic group on its [coordinate ring](/source/Coordinate_ring), it can be shown that every affine algebraic group is a linear (or matrix) group, meaning that it is isomorphic to an algebraic subgroup of the general linear group.

For example, the additive group can be embedded in G L 2 {\displaystyle \mathrm {GL} _{2}} by the morphism x ↦ ( 1 x 0 1 ) {\displaystyle x\mapsto \left({\begin{matrix}1&x\\0&1\end{matrix}}\right)} .

There are many examples of such groups beyond those given previously, including [orthogonal groups](/source/Orthogonal_group), [symplectic groups](/source/Symplectic_group), [unipotent groups](/source/Unipotent_group), [algebraic tori](/source/Algebraic_torus), and certain [semidirect products](/source/Semidirect_product),[7] such as [jet groups](/source/Jet_group), or some [solvable groups](/source/Solvable_group) such as that of invertible [triangular matrices](/source/Triangular_matrix).

Linear algebraic groups can be classified to a certain extent. [Levi's theorem](/source/Levi's_theorem) states that every linear algebraic group is (essentially) a semidirect product of a unipotent group (its [unipotent radical](/source/Unipotent_radical)) with a [reductive group](/source/Reductive_group). In turn, a reductive group is decomposed as (again essentially) a product of its [center](/source/Center_(group_theory)) (an algebraic torus) with a [semisimple group](/source/Semisimple_group). The latter are classified over algebraically closed fields via their [Lie algebras](/source/Semisimple_Lie_algebra#Classification).[8] The classification over arbitrary fields is more involved, but still well-understood.[9] If can be made very explicit in some cases, such as over the real or [*p*-adic](/source/P-adic) fields, and thereby over [number fields](/source/Number_field) via [local-global principles](/source/Local-global_principle).

## Abelian varieties

Main article: [Abelian variety](/source/Abelian_variety)

Abelian varieties are connected projective algebraic groups, such as elliptic curves. They are always commutative. They arise naturally in various situations in algebraic geometry and number theory, such as the [Jacobian varieties](/source/Jacobian_variety) of curves.

## Structure theorem for general algebraic groups

Not all algebraic groups are linear groups or abelian varieties; for instance, some [group schemes](/source/Group_scheme) occurring naturally in [arithmetic geometry](/source/Arithmetic_geometry) are neither.[10] [Chevalley's structure theorem](/source/Chevalley's_structure_theorem) asserts that every connected algebraic group is an extension of an [abelian variety](/source/Abelian_variety) by a [linear algebraic group](/source/Linear_algebraic_group). More precisely, if *K* is a [perfect field](/source/Perfect_field), and *G* a connected algebraic group over *K*, then there exists a unique normal closed subgroup *H* in *G*, such that *H* is a connected linear algebraic group and *G*/*H* an abelian variety.

## Connectedness

As an algebraic variety, G {\displaystyle \mathrm {G} } carries a [Zariski topology](/source/Zariski_topology). It is not in general a [group topology](/source/Topological_group); that is, the group operations may not be continuous for this topology (because the Zariski topology on the product is not the product of Zariski topologies on the factors[11]).

An algebraic group is said to be *connected* if the underlying algebraic variety is connected for the Zariski topology. For an algebraic group, this means that it is not the union of two proper algebraic subsets.[12]

Examples of groups that are not connected are given by the algebraic subgroup of n {\displaystyle n} th [roots of unity](/source/Root_of_unity) in the multiplicative group G m {\displaystyle \mathrm {G} _{m}} (each point is a Zariski-closed subset so it is not connected for n ≥ 1 {\displaystyle n\geq 1} ). This group is generally denoted by μ n {\displaystyle \mu _{n}} . Other non-connected groups are the orthogonal group in even dimension (the determinant gives a surjective morphism to μ 2 {\displaystyle \mu _{2}} ).

More generally, every finite group is an algebraic group (it can be realised as a finite, hence Zariski-closed, subgroup of some G L n {\displaystyle \mathrm {GL} _{n}} by [Cayley's theorem](/source/Cayley's_theorem)). In addition it is both affine and projective. Thus, in particular for classification purposes, it is natural to restrict statements to connected algebraic groups.

## Algebraic groups over local fields and Lie groups

If the field k {\displaystyle k} is a [local field](/source/Local_field) (for instance the real or complex numbers, or a *p*-adic field) and G {\displaystyle \mathrm {G} } is a k {\displaystyle k} -group, then the group G ( k ) {\displaystyle \mathrm {G} (k)} is endowed with the analytic topology coming from any embedding into a projective space P n ( k ) {\displaystyle \mathbb {P} ^{n}(k)} as a quasi-projective variety. This is a group topology, and it makes G ( k ) {\displaystyle \mathrm {G} (k)} into a [topological group](/source/Topological_group). Such groups are important examples in the general theory of topological groups.

If k = R {\displaystyle k=\mathbb {R} } or C {\displaystyle \mathbb {C} } , then this makes G ( k ) {\displaystyle \mathrm {G} (k)} into a [Lie group](/source/Lie_group). Not all Lie groups can be obtained via this procedure; for example, the universal cover of [SL2(**R**)](/source/SL2(R)), or the quotient of the [Heisenberg group](/source/Heisenberg_group) by an infinite normal discrete subgroup.[13] An algebraic group over the real or complex numbers may have closed subgroups (in the analytic topology) that do not have the same connected component of the identity as any algebraic subgroup.

## Coxeter groups and algebraic groups

Further information: [Field with one element](/source/Field_with_one_element)

There are a number of analogous results between algebraic groups and [Coxeter groups](/source/Coxeter_group) – for instance, the number of elements of the symmetric group is n ! {\displaystyle n!} , and the number of elements of the general linear group over a finite field is (up to some factor) the [*q*-factorial](/source/Q-factorial) [ n ] q ! {\displaystyle [n]_{q}!} ; thus, the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the [field with one element](/source/Field_with_one_element), which considers Coxeter groups to be simple algebraic groups over the field with one element.

## See also

- [Character variety](/source/Character_variety)

- [Borel subgroup](/source/Borel_subgroup)

- [Tame group](/source/Tame_group)

- [Morley rank](/source/Morley_rank)

- [Cherlin–Zilber conjecture](/source/Cherlin%E2%80%93Zilber_conjecture)

- [Adelic algebraic group](/source/Adelic_algebraic_group)

- [Pseudo-reductive group](/source/Pseudo-reductive_group)

## References

1. **[^](#cite_ref-FOOTNOTEBorel1991p.54_1-0)** [Borel 1991](#CITEREFBorel1991), p.54.

1. **[^](#cite_ref-FOOTNOTEBorel1991p._46_2-0)** [Borel 1991](#CITEREFBorel1991), p. 46.

1. **[^](#cite_ref-FOOTNOTEBorel19911.6(2),_p._49_3-0)** [Borel 1991](#CITEREFBorel1991), 1.6(2), p. 49.

1. **[^](#cite_ref-FOOTNOTEBorel1991Corollary_1.4,_p._47_4-0)** [Borel 1991](#CITEREFBorel1991), Corollary 1.4, p. 47.

1. **[^](#cite_ref-FOOTNOTEBorel1991Theorem_6.8,_p._98_5-0)** [Borel 1991](#CITEREFBorel1991), Theorem 6.8, p. 98.

1. **[^](#cite_ref-FOOTNOTEBorel19913.5,_p._65_6-0)** [Borel 1991](#CITEREFBorel1991), 3.5, p. 65.

1. **[^](#cite_ref-FOOTNOTEBorel1991pp._55-56_7-0)** [Borel 1991](#CITEREFBorel1991), pp. 55-56.

1. **[^](#cite_ref-FOOTNOTEBorel199124.1_8-0)** [Borel 1991](#CITEREFBorel1991), 24.1.

1. **[^](#cite_ref-FOOTNOTEBorel199124.2_9-0)** [Borel 1991](#CITEREFBorel1991), 24.2.

1. **[^](#cite_ref-10)** Conrad, Brian (2002). "A modern proof of Chevalley's theorem on algebraic groups". *J. Ramanujan Math. Soc*. **17** (1): 1–18. [Zbl](/source/Zbl_(identifier)) [1007.14005](https://zbmath.org/?format=complete&q=an:1007.14005).

1. **[^](#cite_ref-FOOTNOTEBorel1991p._16_11-0)** [Borel 1991](#CITEREFBorel1991), p. 16.

1. **[^](#cite_ref-FOOTNOTEBorel1991p._47_12-0)** [Borel 1991](#CITEREFBorel1991), p. 47.

1. **[^](#cite_ref-13)** ["Non-linear Lie group"](https://mathoverflow.net/questions/91789/non-linear-lie-group). *MathOverflow*. Retrieved May 13, 2022.

- Chevalley, Claude, ed. (1958), [*Séminaire C. Chevalley, 1956--1958. Classification des groupes de Lie algébriques*](https://web.archive.org/web/20141104160713/http://www.numdam.org/numdam-bin/browse?id=SCC_1956-1958__1_), 2 vols, Paris: Secrétariat Mathématique, [MR](/source/MR_(identifier)) [0106966](https://mathscinet.ams.org/mathscinet-getitem?mr=0106966), Reprinted as volume 3 of Chevalley's collected works., archived from [the original](http://www.numdam.org/numdam-bin/browse?id=SCC_1956-1958__1_) on 2014-11-04, retrieved 2012-06-25

- [Borel, Armand](/source/Armand_Borel) (1991). *Linear algebraic groups. 2nd enlarged ed*. Graduate Texts in Mathematics. Springer-Verlag. pp. x+288. [Zbl](/source/Zbl_(identifier)) [0726.20030](https://zbmath.org/?format=complete&q=an:0726.20030).

- Humphreys, James E. (1972), *Linear Algebraic Groups*, [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), vol. 21, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [ISBN](/source/ISBN_(identifier)) [978-0-387-90108-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90108-4), [MR](/source/MR_(identifier)) [0396773](https://mathscinet.ams.org/mathscinet-getitem?mr=0396773)

- [Lang, Serge](/source/Serge_Lang) (1983), *Abelian varieties*, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [ISBN](/source/ISBN_(identifier)) [978-0-387-90875-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90875-5)

- [Milne, J. S.](/source/James_Milne_(mathematician)) (2017), *Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field*, [Cambridge University Press](/source/Cambridge_University_Press), [doi](/source/Doi_(identifier)):[10.1017/9781316711736](https://doi.org/10.1017%2F9781316711736), [ISBN](/source/ISBN_(identifier)) [978-1107167483](https://en.wikipedia.org/wiki/Special:BookSources/978-1107167483), [MR](/source/MR_(identifier)) [3729270](https://mathscinet.ams.org/mathscinet-getitem?mr=3729270)

- Milne, J. S., *[Affine Group Schemes; Lie Algebras; Lie Groups; Reductive Groups; Arithmetic Subgroups](http://www.jmilne.org/math/CourseNotes/ala.html)*

- [Mumford, David](/source/David_Mumford) (1970), *Abelian varieties*, [Oxford University Press](/source/Oxford_University_Press), [ISBN](/source/ISBN_(identifier)) [978-0-19-560528-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-560528-0), [OCLC](/source/OCLC_(identifier)) [138290](https://search.worldcat.org/oclc/138290)

- Springer, Tonny A. (1998), *Linear algebraic groups*, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston, [ISBN](/source/ISBN_(identifier)) [978-0-8176-4021-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8176-4021-7), [MR](/source/MR_(identifier)) [1642713](https://mathscinet.ams.org/mathscinet-getitem?mr=1642713)

- [Waterhouse, William C.](/source/William_C._Waterhouse) (1979), [*Introduction to affine group schemes*](https://books.google.com/books?id=l0DgAIx_djoC&q=%22algebraic+group%22), Graduate Texts in Mathematics, vol. 66, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [ISBN](/source/ISBN_(identifier)) [978-0-387-90421-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-90421-4)

- [Weil, André](/source/Andr%C3%A9_Weil) (1971), *Courbes algébriques et variétés abéliennes*, Paris: Hermann, [OCLC](/source/OCLC_(identifier)) [322901](https://search.worldcat.org/oclc/322901)

## Further reading

- [Algebraic groups and their Lie algebras](https://github.com/dkmiller/algebraic-groups/releases/download/refs%2Fheads%2Fmaster/algebraic-groups.pdf) by Daniel Miller

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Adapted from the Wikipedia article [Algebraic group](https://en.wikipedia.org/wiki/Algebraic_group) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Algebraic_group?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.