nLab algebraic group

Definition

Given a (typically algebraically closed) field $k$, an algebraic $k$-group is a group object in the category of $k$-varieties.

Linear algebraic groups and abelian varieties

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 $H$ such that their quotient $G/H$ is an abelian variety.

Linear algebraic group

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

An algebraic group is linear iff it is affine.

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

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

Abelian variety

Another important class are connected algebraic $k$-groups whose underlying variety is projective; these are automatically commutative so they are called 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.

Elliptic curve

An abelian variety of dimension $1$ is called an elliptic curve.

Other prominent classes of algebraic groups

Some of the definitions of the following classes exist more generally for group schemes.

Jacobian

(…)

Unipotent algebraic groups

(See also more generally unipotent group scheme.)

Definition

An element $x$ of an affine algebraic group is called unipotent if its associated right translation operator $r_x$ on the affine coordinate ring? $A[G]$ of $G$ is locally unipotent as an element of the ring of linear endomorphism of $A[G]$ where ‘’locally unipotent’‘ means that its restriction to any finite dimensional stable subspace of $A[G]$ is unipotent as a ring object.

Theorem

(Jordan-Chevalley decomposition?) Any commutative linear algebraic group over a perfect field is the product of a unipotent and a semisimple algebraic group.

Properties

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.

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

Examples

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

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

Generalizations

• group scheme

Related concepts

• isogeny

• form of an algebraic group

References

On algebraic schemes and algebraic groups via functorial geometry:

• Michel Demazure, Pierre Gabriel, Groupes algebriques, tome 1, avec un appendice Corps de classes local par Hazewinkel M, Mason and Cie, Paris (1970)

  (later volumes never appeared)

English translation:

• Michel Demazure, Peter Gabriel, Introduction to algebraic geometry and algebraic groups, North-Holland mathematics studies 39 (1980) [ISBN:9780080871509]

See also:

• Gerhard P. Hochschild, Basic Theory of Algebraic Groups and Lie Algebras, Graduate Texts in Mathematics 75, Springer (1981) [doi:10.1007/978-1-4613-8114-3_16]

• M. Artin, J. E. Bertin, M. Demazure, P. Gabriel, A. Grothendieck, M. Raynaud, J.-P. Serre, Schemas en groupes, SGA3

• James Milne, Algebraic Groups – The theory of group schemes of finite type over a field, Cambridge University Press (2017) [doi:10.1017/9781316711736, webpage, pdf]

• Armand Borel, Linear algebraic groups, Springer (2nd edition much expanded)

• W. Waterhouse, Introduction to affine group schemes, GTM 66, Springer 1979.

• Serge Lang, Abelian varieties, Springer 1983.

• David 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)

• Roe Goodman, Nolan R. Wallach, Symmetry, representations, and invariants. Graduate Texts in Mathematics, 255. Springer, Dordrecht, 2009.

• J. Milne, Algebraic groups, Lie Groups, and their arithmetic subgroups, pdf