derived smooth geometry
There are two important classes of algebraic groups whose intersection is trivial (the identity group): Linear algebraic groups and abelian varieties.
Any algebraic group contains a unique normal linear algebraic subgroup such that their quotient is an abelian variety.
An algebraic group is linear iff it is affine.
An algebraic group scheme is affine if the underlying scheme is affine.
Another important class are connected algebraic -groups whose underlying variety is projective; these are automatically commutative so they are called abelian varieties. In dimension these are precisely the elliptic curves. If is a perfect field and an algebraic -group, the theorem of Chevalley says that there is a unique linear subgroup such that is an abelian variety.
An abelian variety of dimension is called an elliptic curve.
Some of the definitions of the following classes exist more generally for group schemes.
(See also more generally unipotent group scheme.)
An element of an affine algebraic group is called unipotent if its associated right translation operator on the affine coordinate ring? of is locally unipotent as an element of the ring of linear endomorphism of where ‘’locally unipotent’‘ means that its restriction to any finite dimensional stable subspace of is unipotent as a ring object.
Among group schemes are ‘the infinite-dimensional algebraic groups’ of Shafarevich.
The affine line without its origin, comes canonically with the structure of a group under multiplication: the multiplicative group .
The standard references are
M. Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970
SGA3 Schémas en groupes, 1962–1964, Lecture Notes in Mathematics 151, 152 and 153, 1970
M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, Schemas en groupes, i.e. SGA III-1, III-2, III-3
A. Borel, Linear algebraic groups, Springer (2nd edition much expanded)
W. Waterhouse, Introduction to affine group schemes, GTM 66, Springer 1979.
S. Lang, Abelian varieties, Springer 1983.
D. Mumford, Abelian varieties, 1970, 1985.
J. C. Jantzen, Representations of algebraic groups, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)
T. Springer, Linear algebraic groups, Progress in Mathematics 9, Birkhäuser Boston (2nd ed. 1998, reprinted 2008)