group variety

Let $G$ be a variety  (an affine (http://planetmath.org/AffineVariety), projective (http://planetmath.org/ProjectiveVariety), or quasi-projective variety). We say $G$ is a group variety if $G$ is provided with morphisms of varieties:

 $\displaystyle\mu:G\times G$ $\displaystyle\to G$ $\displaystyle(g_{1},g_{2})$ $\displaystyle\mapsto g_{1}g_{2},$
 $\displaystyle\iota:G$ $\displaystyle\to G$ $\displaystyle g$ $\displaystyle\mapsto g^{-1},$

and

 $\displaystyle\epsilon:\{*\}$ $\displaystyle\to G$ $\displaystyle\mapsto e,$

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

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 (http://planetmath.org/VarietyOfGroups)”.

Title group variety GroupVariety 2013-03-22 14:09:37 2013-03-22 14:09:37 archibal (4430) archibal (4430) 4 archibal (4430) Definition msc 14L10 msc 14K99 msc 20G15 AffineAlgebraicGroup GroupScheme VarietyOfGroups