group variety

Let G be a varietyPlanetmathPlanetmath (an affine (, projective (, or quasi-projective variety). We say G is a group variety if G is provided with morphisms of varieties:

μ:G×G G
(g1,g2) g1g2,
ι:G G
g g-1,


ϵ:{*} G

and if these morphisms make the elements of G into a group.

In short, G should be a group object in the category of varieties. Examples include the general linear groupMathworldPlanetmath of dimension n on k and elliptic curvesMathworldPlanetmath.

Group varieties that are actually projective are in fact abelian groupsMathworldPlanetmath (although this is not obvious) and are called abelian varietiesMathworldPlanetmath; their study is of interest to number theorists (among others).

Just as schemes generalize varieties, group schemes generalize group varieties. When dealing with situations in positive characteristic, or with families of group varieties, often they are more appropriate.

There is also a (not very closely related) concept in group theory of a “variety of groups (”.

Title group variety
Canonical name GroupVariety
Date of creation 2013-03-22 14:09:37
Last modified on 2013-03-22 14:09:37
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 4
Author archibal (4430)
Entry type Definition
Classification msc 14L10
Classification msc 14K99
Classification msc 20G15
Related topic AffineAlgebraicGroup
Related topic GroupScheme
Related topic VarietyOfGroups