Algebraic group
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, including 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.
Definitions
Formally, an algebraic group over a field is an algebraic variety over, together with a distinguished element , and regular maps and that satisfy the group axioms.Examples
- The additive group: the affine line endowed with addition and opposite as group operations is an algebraic group. It is called the additive group, and usually denoted by.
- The multiplicative group: Let be the affine variety defined by the equation in the affine plane. The functions and are regular on, and they satisfy the group axioms. The algebraic group is called the multiplicative group, because its -points are isomorphic to the multiplicative group of the field .
- The special linear group is an algebraic group: it is given by the algebraic equation in the affine space , multiplication of matrices is regular and the formula for the inverse in terms of the adjugate matrix shows that inversion is regular as well on matrices with determinant 1.
- The general linear group of invertible matrices over a field is an algebraic group. It can be realized as a subvariety in in much the same way as the multiplicative group in the previous example.
- A non-singular cubic curve in the projective plane with a specified point can be endowed with a geometrically defined group law that makes it into an algebraic group.
Related definitions
A morphism between two algebraic groups is a regular map that is also a group homomorphism. Its kernel is an algebraic subgroup of, and 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. 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, then the morphism of groups may not be surjective.
Lie algebra of an algebraic group
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.Alternative definitions
A more sophisticated definition of an algebraic group over a field is that it is a group scheme over .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.
Affine algebraic groups
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, it can be shown that every affine algebraic group is a linear 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, including orthogonal groups, symplectic groups, unipotent groups, algebraic tori, and certain semidirect products, such as jet groups, or some solvable groups such as that of invertible triangular matrices.
Linear algebraic groups can be classified to a certain extent. Levi's theorem states that every linear algebraic group is a semidirect product of a unipotent group with a reductive group. In turn, a reductive group is decomposed as a product of its center with a semisimple group. The latter are classified over algebraically closed fields via their Lie algebras. The classification over arbitrary fields is more involved, but still well-understood. If can be made very explicit in some cases, such as over the real or p-adic fields, and thereby over number fields via local-global principles.
Abelian varieties
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 of curves.Structure theorem for general algebraic groups
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, 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, carries a Zariski topology. It is not in general a group topology; that is, the group operations may not be continuous for this topology.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 th roots of unity in the multiplicative group . This group is generally denoted by. Other non-connected groups are the orthogonal group in even dimension.
More generally, every finite group is an algebraic group. 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 is a local 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, 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 that do not have the same connected component of the identity as any algebraic subgroup.