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algebraic group

Definition

Given a field $k$, an algebraic $k$-group is a group object in the category of $k$-varieties.

Special cases

An algebraic $k$-group is linear if it is a Zariski-closed subgroup of the general linear group $\mathrm{GL}(n,k)$ for some $n$.

Another important class are commutative algebraic $k$-groups whose underlying variety is projective, namely the abelian varieties; in dimension $1$ these are precisely the elliptic curves. If $k$ is a perfect field and $G$ an algebraic $k$-group, the theorem of Chevalley says that there is a unique linear subgroup $H\subset G$ such that $G/H$ is an abelian variety.

The group objects in the category of algebraic schemes and formal schemes are called (algebraic) group schemes and formal groups, respectively.

Among group schemes are ‘the infinite-dimensional algebraic groups’ of Shafarevich.

An algebraic group scheme is affine if the underlying scheme is affine.

Algebraic analogues of loop groups are in the category of ind-schemes. All linear algebraic $k$-groups are affine.

The category of affine group schemes is the opposite of the category of commutative Hopf algebras.

Concrete examples

The affine line ${𝔸}^1$ comes canonically with the structure of a group under addtion: the additive group $\mathrm{mathbb}{G}_a$.

The affine line without its origin, ${𝔸}^1-\{0\}$ comes canonically with the structure of a group under multiplication: the multiplicative group ${𝔾}_m$.

References

The standard references are

• M. Demazure, P. Gabriel, Groupes algebriques, tome 1 (later volumes never appeared), Mason and Cie, Paris 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)