In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties.
An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called linear algebraic groups. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states that every algebraic group can be constructed from groups in those two families.
Formally, an algebraic group over a field
k
G
k
e\inG(k)
G x G\toG
G\toG
A1
k
k
Ga
Gm
xy=1
A2
((x,y),(x',y'))\mapsto(xx',yy')
(x,y)\mapsto(x-1,y-1)
Gm
(1,1)
Gm
k
k
x\mapsto(x,x-1)
A1
SLn
\det(g)=1
n2 | |
A |
n
n
GLn
k
n2+1 | |
A |
P2
An algebraic subgroup of an algebraic group
G
H
G
G
G x G\toG
G\toG
H x H
H
H
A morphism between two algebraic groups
G,G'
G\toG'
G
G'
Quotients 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 (which are regular maps). If
H
G
G/H
\pi:G\toG/H
H
\pi
k
G(k)\toG(k)/H(k)
Similarly to the Lie group–Lie algebra correspondence, to an algebraic group over a field
k
k
A more sophisticated definition of an algebraic group over a field
k
k
Yet another definition of the concept is to say that an algebraic group over
k
k
See main article: 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 groups and the general and special linear groups are affine. Using the action of an affine algebraic group on its 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
GL2
x\mapsto\left(\begin{smallmatrix}1&x\ 0&1\end{smallmatrix}\right)
There are many examples of such groups beyond those given previously:
Linear algebraic groups can be classified to a certain extent. Levi's theorem states that every such is (essentially) a semidirect product of a unipotent group (its unipotent radical) with a reductive group. In turn reductive groups are decomposed as (again essentially) a product of their center (an algebraic torus) with a semisimple group. The latter are classified over algebraically closed fields via their Lie algebra. The classification over arbitrary fields is more involved but still well-understood. If can be made very explicit in some cases, for example over the real or p-adic fields, and thereby over number fields via local-global principles.
See main article: Abelian variety.
Abelian varieties are connected projective algebraic groups, for instance elliptic curves. They are always commutative. They arise naturally in various situations in algebraic geometry and number theory, for example as the Jacobian variety of a curve.
Not all algebraic groups are linear groups or abelian varieties, for instance some group schemes occurring naturally in arithmetic geometry are neither.[1] Chevalley's structure theorem asserts that every connected algebraic group is an extension of an abelian variety by a linear algebraic group. More precisely, if K is a perfect field, and G a connected algebraic group over K, 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.
As an algebraic variety
G
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.
Examples of groups that are not connected are given by the algebraic subgroup of
n
Gm
n\ge1
\mun
\mu2
More generally every finite group is an algebraic group (it can be realised as a finite, hence Zariski-closed, subgroup of some
GLn
If the field
k
G
k
G(k)
Pn(k)
G(k)
If
k=R
C
G(k)
There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is
n!
[n]q!