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algebraic group

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Definition

Given a (typically algebraically closed) field kk, an algebraic kk-group is a group object in the category of kk-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 HH such that their quotient G/HG/H is an abelian variety.

Linear algebraic group

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

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 commutative algebraic kk-groups whose underlying variety is projective, namely the abelian varieties; in dimension 11 these are precisely the elliptic curves. If kk is a perfect field and GG an algebraic kk-group, the theorem of Chevalley says that there is a unique linear subgroup HGH\subset G such that G/HG/H is an abelian variety.

Elliptic curve

An abelian variety of dimension 11 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 xx of an affine algebraic group is called unipotent if its associated right translation operator r xr_x on the affine coordinate ring? A[G]A[G] of GG is locally unipotent as an element of the ring of linear endomorphism of A[G]A[G] where ”locally unipotent” means that its restriction to any finite dimensional stable subspace of A[G]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 kk-groups are affine.

Examples

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

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

Generalizations

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)

Revised on July 17, 2013 11:23:21 by Urs Schreiber (82.169.65.155)