Algebraic group

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.[1]. 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.

There are other algebraic groups, but Chevalley's structure theorem asserts that every algebraic group is an extension of an abelian variety by a linear algebraic group. More precisely, if K is a perfect field, and G an algebraic group over K, there exists a unique normal closed subgroup H in G, such that H is a linear algebraic group and G/H an abelian variety.

According to another basic theorem[which?], any group that is also an affine variety has a faithful finite-dimensional linear representation: it is isomorphic to a matrix group, defined by polynomial equations.

Over the fields of real and complex numbers, every algebraic group is also a Lie group, but the converse is false.

A group scheme is a generalization of an algebraic group that allows, in particular, working over a commutative ring instead of a field.

An algebraic subgroup of an algebraic group is a Zariski-closed subgroup. Generally these are taken to be connected (or irreducible as a variety) as well.

Another way of expressing the condition is as a subgroup that is also a subvariety.

This may also be generalized by allowing schemes in place of varieties. The main effect of this in practice, apart from allowing subgroups in which the connected component is of finite index > 1, is to admit non-reduced schemes, in characteristic p.

There are a number of mathematical notions to study and classify algebraic groups.