In mathematics, an algebraic group is an algebraic variety endowed with a group structure which 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 is an algebraic variety over , together with a distinguished element (the neutral element), and regular maps (the multiplication operation) and (the inversion operation) which satisfy the group axioms.
An algebraic subgroup of an algebraic group is a subvariety of which is also a subgroup of (that is, the maps and defining the group structure map and , respectively, into ).
A morphism between two algebraic groups is a regular map which is also a group morphism. Its kernel is an algebraic sugroup of , its image is an algebraic subgroup of .
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 is a normal algebraic subgroup of then there exists an algebraic group and a surjective morphism such that is the kernel of . Note that if the field is not algebraically closed, the morphism of groups may not be surjective (the default of surjectivity is measured by Galois cohomology).
Similarly to the Lie group–Lie algebra correspondence, to an algebraic group over a field is associated a Lie algebra over . As a vector space the Lie algebra is isomorphic to the tangent space at the identity element. The Lie bracket can be constructed from its interpretation as a space of derivations.
A more sophisticated definition of an algebraic group over a field is that it is that of a group scheme over (group schemes can more generally be defined over commutative rings).
Yet another definition of the concept is to say that an algebraic group over is a group object in the category of algebraic varieties over .
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 by the morphism .
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.
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. 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 carries a Zariski topology. It is not in general a group topology, i.e. the group operations may not be continuous for this topology (because Zariski topology on the product is not the product of Zariski topologies on the factors).
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 which are not connected are given by the algebraic subgroup of th roots of unity in the multiplicative group (each point is a Zariski-closed subset so it is not connected for ). This group is generally denoted by . Another non-connected group are orthogonal group in even dimension (the determinant gives a surjective morphism to ).
More generally every finite group is an algebraic group (it can be realised as a finite, hence Zariski-closed, subgroup of some by 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 group.
If the field is a local field (for instance the real or complex numbers, or a p-adic field) and is a -group then the group is endowed with the analytic topology coming from any embedding into a projective space as a quasi-projective variety. This is a group topology, and it makes into a topological group. Such groups are important examples in the general theory of topological groups.
If or then this makes into a Lie group. Not all Lie groups can be obtained via this procedure, for example the universal cover of SL2(R), or the quotient of the Heisenberg group by an infinite normal discrete subgroup. An algebraic group over the real or complex numbers may have closed subgroups (in the analytic topology) which do not have the same connected component of the identity as any algebraic subgroup.
There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is , and the number of elements of the general linear group over a finite field is (up to some factor) the q-factorial ; 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, which considers Coxeter groups to be simple algebraic groups over the field with one element.