Algebraic group

Abstract definition
An algebraic group or group variety over a field $$K$$ is a defining ingredient::group object in the category of algebraic varieties over $$K$$.

Concrete definition
An algebraic group or group variety over a field $$K$$ is a set $$G$$ equipped with the following two structures:


 * The structure of an defining ingredient::algebraic variety over the defining ingredient::field $$K$$.
 * The structure of a defining ingredient::group

such that the following compatibility conditions are satisfied:

Variation using separately instead of jointly algebraic
For an algebraic variety $$V$$, a map $$f: V \times V \to V$$ is termed separately algebraic if, for any fixed $$v \in V$$, the maps $$x \mapsto f(x,v)$$ and $$x \mapsto f(v,x)$$ are both regular morphisms from $$V$$ to itself. This is in contrast with jointly algebraic, which means regular from $$V \times V$$ with the product variety structure to $$V$$.

For infinite countable fields, it is possible to construct separately algebraic maps that are not jointly algebraic, even in the case where the variety $$V$$ is the field itself. Thus, a priori, we could define a separately algebraic version of the algebraic group definition where we require the group multiplication to only be separately algebraic, while still requiring the inverse map to be continuous. However, it turns out that any separately algebraic group must be algebraic.

However, we do obtain separate notions of left-algebraic group and right-algebraic group, where we assume algebraicity in only one of the two coordinates.

Note on algebraic closure assumptions
The definition of algebraic group makes sense over any field. However, because many of the foundational results of algebraic geometry are valid only for algebraically closed fields, most basic treatments of algebraic groups consider algebraic groups over algebraically closed fields. To study what happens over a non-algebraically closed field, we typically put it inside its algebraic closure, then consider the algebraic group over the algebraic closure, and, if the formulas defining the group operations are all realized over the subfield, we can look at the points of the algebraic group corresponding to that subfield. For instance, to study elliptic curve groups over $$\mathbb{Q}$$, we consider the corresponding elliptic curve group over $$\mathbb{C}$$, then consider the subgroup of rational points inside that elliptic curve.

Flattened local structure near the identity

 * Formal group law of an algebraic group: This depends only on the connected component of identity.
 * Lie algebra of an algebraic group: This depends only on the connected component of identity, and further, is invariant under finite degree coverings (or something like that).

Topological structure
The algebraic variety underlying an algebraic group has a topology. With this topology, it becomes a quasitopological group but almost never becomes a topological group. The reason is that the multiplication map is not jointly continuous with respect to the product topology on the product. This is related to the fact that the product topology on the product is considerably coarser than the Zariski topology on the product. An algebraic group is a quasitopological group because the multiplication map is separately continuous and the inverse map is continuous.

Moreover, in the finite-dimensional case, the quasitopological group is compact, sober, and T1 (hence, a T0 quasitopological group). See:


 * Algebraic group implies quasitopological group
 * Algebraic group not implies topological group

Topological and analytic structure for some fields
In case the underlying field is a topological field, algebraic groups do become topological groups, not with respect to the Zariski topology, but with respect to a topology arising from the topology of the field. In fact, for fields that have an analytic structure, such as the field of real numbers, field of complex numbers, and various extensions of the p-adics, we get an analytic structure on the algebraic group, which is called a Lie group structure.

Linear algebraic groups and abelian varieties
There are two extreme cases of algebraic groups over algebraically closed fields: linear algebraic groups and abelian varieties. Every algebraic group can be broken down in terms of groups of these sorts:


 * Any algebraic group has a unique closed normal linear algebraic subgroup so that the quotient group is an abelian variety

Relation with Lie group structure
For fields such as the real and complex numbers that have an analytic structure, algebraic groups are Lie groups. Some facts:


 * Quotient map of Lie group structures for algebraic groups need not be quotient map of algebraic groups
 * Closed normal subgroup for Lie structure need not be closed normal subgroup of algebraic group