Algebraic group

In algebraic geometry, an algebraic group (or group variety) is a group (mathematics) that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. In category theory terms, an algebraic group is a group object in the category (mathematics) of algebraic variety.

Several important classes of groups are algebraic groups, including:

Finite groups

GL n C, the general linear group of invertible matrices over C

Elliptic curves

Two important classes of algebraic groups arise, that for the most part are studied separately: abelian variety (the projective theory) and linear algebraic groups (the affine theory). There are certainly examples that are neither one nor the other — these occur for example in the modern theory of differential of the first kind such as the Weierstrass zeta function, or the theory of generalized Jacobians. But according to a basic theorem the general algebraic group is a semidirect product of an abelian variety with a linear algebraic group.

According to another basic theorem, any group in the category of of an affine algebraic group G is necessarily of finite index in G .

When one wants to work over a base ring R (commutative), there is the . There is quite a refined theory of group schemes, that enters for example in the contemporary theory of abelian varieties.