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; 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) that satisfy the group axioms. The additive group: the affine line endowed with addition and opposite as group operations is an algebraic group. It is called the additive group (because its -points are isomorphic as a group to the additive group of ), 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 (with neutral element ). The algebraic group is called multiplicative group, because its -points are isomorphic to the multiplicative group of the field (an isomorphism is given by ; note that the subset of invertible elements does not define an algebraic subvariety in ).