User:Jean Raimbault/sandbox/Algebraic group

Definition
Let $$k$$ be a field; a $$k$$-algebraic group or algebraic group over $$k$$ is an algebraic variety $$G$$ defined over $$k$$ together with a regular map (also defined over $$k$$) $$m : G \times G \to G$$ (a multiplication map) satisfying the following diagram:

image

(this is the algebro-geometric version of associativity) and such that there exists two regular maps $$e : * \to G $$ (where $$* = \mathrm{Spec}(k)$$ is a point) and $$ \mathrm{inv} : G \to G$$ satisfying the following diagrams:

images

which correspond to the classical notions of neutral element and inverse map.

A morphism of algebraic groups is a regular map which intertwines the multiplications maps, i.e. if $$G, G'$$ are algebraic groups with multiplication maps $$m, m'$$ and $$\phi: G \to H$$ is a regular map between the underlying varieties, it is a morphism if and only if $$m' \circ (\phi \times \phi) = \phi \circ m$$.

The notion thus defined depends somewhat on the notion of algebraic variety in use. In classical references it is usually understood in the sense of Weil's Foundations In modern terminology, using Grothendieck's theory of schemes, algebraic groups are group schemes over a field, with the additional requirement that the underlying scheme be a variety.(Springer, Milne) It can be interesting to admit more general group schemes, including those admitting nilpotent elements in their local rings (ex. below).

Examples

 * Finite groups
 * GL(n, C), the general linear group of invertible matrices over C
 * Jet group
 * Elliptic curves.

Non-examples: PSL2(R), revêtement dble de SL2R SL2(Z) but group scheme over spec(Z)

The Zariski topology and algebraic subgroups
alg subgroups == Z-closed sbgps

Classes of algebraic groups
The study of algebraic groups is essentially divided into that of two subclasses: 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.
 * Abelian varieties are algebraic groups whose underlying algebraic variety is projective (for example elliptic curves); those groups are always abelian
 * Linear algebraic groups are algebraic groups whose underlying algebraic variety is affine; equivalently, they are those groups which admit can be embedded into a group GLn for some n''.

Linear algebraic groups

 * diagonalisable/torus
 * unipotent
 * semisimple

Finite Coxeter groups and algebraic groups over finite fields
groups of Lie type blabla

There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is $$n!$$, and the number of elements of the general linear group over a finite field is the q-factorial $$[n]_q!$$; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element.