Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties.
Formally, an algebraic group over a field
(the neutral element), and regular maps
(the inversion operation) that satisfy the group axioms.
defining the group structure map
[4] 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).
is not algebraically closed, the morphism of groups
The Lie bracket can be constructed from its interpretation as a space of derivations.
[6] A more sophisticated definition of an algebraic group over a field
(group schemes can more generally be defined over commutative rings).
is a group object in the category of algebraic varieties over
For example the additive group can be embedded in
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.
[8] The classification over arbitrary fields is more involved but still well-understood.
[9] 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 arise naturally in various situations in algebraic geometry and number theory, for example as the Jacobian variety of a curve.
[10] 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.
[12] Examples of groups that 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
Thus, in particular for classification purposes, it is natural to restrict statements to connected algebraic group.
is endowed with the analytic topology coming from any embedding into a projective space
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.
[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.
, and the number of elements of the general linear group over a finite field is (up to some factor) the q-factorial
This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element.