Algebraic group

In algebraic geometry, two important classes of algebraic group arise, that for the most part are studied separately. The general definition of algebraic group is the expected one: a group in the category of algebraic varieties; or, more simply, a group with operations that are described by polynomials over a field K.

It is usual to separate out the theories of abelian varieties (the 'projective' theory) from that of the linear algebraic group[?] (the 'affine' theory). There are certainly examples that are neither one nor the other - these occur for example in the modern theory of integrals of the second and third kinds such as the Weierstrass zeta-function. 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 affine varieties has a faithful linear representation[?]: we can consider it to be a matrix group over K, defined by polynomials over K and with group operation simply matrix multiplication. For that reason a concept of affine algebraic group is redundant over a field - we may as well use a very concrete definition. Note that this means that algebraic group is narrower than Lie group, when working over the field of real numbers: there are examples such as the universal cover of the 2x2 special linear group that are Lie groups, but have no faithful linear representation.

When one wants to work over a base ring R (commutative), there is the group scheme[?] concept: that is, a group in the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra[?]. There is quite a refined theory of group schemes, that enters for example in the contemporary theory of abelian varieties.