Algebraic group

The mathematical notion of algebraic group represents the synthesis of group theory and algebraic geometry represents a key example is the group of invertible n × n matrices.

Definition

An algebraic group is a group object in the category of algebraic varieties over a solid body, ie a variety over a field with

• A morphism (multiplication)
• A morphism ( inverse element )
• And an excellent point ( neutral element ),

So that the following conditions are met:

• Associative law :;
• Neutral element :;
• Inverse element :; here is the inclusion of the diagonal and the Strukturmorphismus.

These conditions are to the requirement that for each schema define equivalent on the set of - valued points the structure of an ( ordinary ) group.

Examples

• The additive group: with the addition as group structure. In particular, for the affine line with the addition.
• The multiplicative group: with multiplication as group structure. In particular, for this is the open subset of the multiplication.
• The general linear group :; it refers to the right side, the group of invertible matrices with entries in the ring. can be identified with.
• The core of a morphism of algebraic groups is an algebraic group again. For example, is an algebraic group.
• Elliptic curves or abelian varieties general.
• Zariski - closed subgroups of algebraic groups are algebraic groups again. Zariski - closed subgroups of linear algebraic groups are as indicated. If an algebraic group is an affine variety, then it is a linear algebraic group.

Set of Chevalley

Every algebraic group over a field of characteristic 0 is ( uniquely ) an extension of an abelian variety by a linear algebraic group. This means that at any algebraic group, there is a maximum linear algebraic subgroup, this is normal and the quotient is an abelian variety:

The picture is the Albanese mapping.