Algebraic Group - Classes

Classes

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 varieties (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 integrals of the second and third kinds such as the Weierstrass zeta function, or the theory of generalized Jacobians. But according to Chevalley's structure theorem any algebraic group is an extension of an abelian variety by a linear algebraic group. This is a result of Claude Chevalley: 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 group and G/H an abelian variety.

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 matrix multiplication as the group operation. 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 2×2 special linear group that are Lie groups, but have no faithful linear representation. A more obvious difference between the two concepts arises because the identity component 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 group scheme concept: that is, a group object 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.