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In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. In category theoretic terms, an algebraic group is a group object in the category of algebraic varieties.

Classes

Several important classes of groups are algebraic groups, including:

Finite groups
GL(n, C), the general linear group of invertible matrices over C
Elliptic curves.

Two important classes of algebraic groups arise, that for the most part are studied separately: abelian varieties (the 'projective' theory) and linear algebraic groups (the 'affine' theory). There are certainly examples that are neither one nor the other — these occur for example in the modern theory of integrals of the second and third kinds such as the Weierstrass zeta function, or the theory of generalized Jacobians. But according to a basic theorem any algebraic group is an extension of an abelian variety by a linear algebraic group. This is a result of Claude Chevalley: if K is a perfect field, and G an algebraic group over K, there exists a unique normal closed subgroup H in G, such that H is a linear group and G/H an abelian variety.[1]

According to another basic theorem, any group in the category of affine varieties has a faithful linear representation: we can consider it to be a matrix group over K, defined by polynomials over K and with matrix multiplication as the group operation. For that reason a concept of affine algebraic group is redundant over a field — we may as well use a very concrete definition. Note that this means that algebraic group is narrower than Lie group, when working over the field of real numbers: there are examples such as the universal cover of the 2×2 special linear group that are Lie groups, but have no faithful linear representation. A more obvious difference between the two concepts arises because the identity component of an affine algebraic group G is necessarily of finite index in G.

When one wants to work over a base ring R (commutative), there is the group scheme concept: that is, a group object in the category of schemes over R. Affine group scheme is the concept dual to a type of Hopf algebra. There is quite a refined theory of group schemes, that enters for example in the contemporary theory of abelian varieties.
Algebraic subgroup

An algebraic subgroup of an algebraic group is a Zariski closed subgroup. Generally these are taken to be connected (or irreducible as a variety) as well.

Another way of expressing the condition is as a subgroup which is also a subvariety.

This may also be generalized by allowing schemes in place of varieties. The main effect of this in practice, apart from allowing subgroups in which the connected component is of finite index > 1, is to admit non-reduced schemes, in characteristic p.
Coxeter groups
Main article: Coxeter group
Further information: Field with one element

There are a number of analogous results between algebraic groups and Coxeter groups – for instance, the number of elements of the symmetric group is n!, and the number of elements of the general linear group over a finite field is the q-factorial [n]_q!; thus the symmetric group behaves as though it were a linear group over "the field with one element". This is formalized by the field with one element, which considers Coxeter groups to be simple algebraic groups over the field with one element.

Algebraic topology (object)
Borel subgroup
Tame group
Morley rank
Cherlin–Zilber conjecture
Glossary of algebraic groups

Notes

^ Chevalley's result is from 1960 and difficult. Contemporary treatment by Brian Conrad: PDF.

References

Humphreys, James E. (1972), Linear Algebraic Groups, Graduate Texts in Mathematics, 21, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90108-4, MR0396773
Lang, Serge (1983), Abelian varieties, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90875-5
Milne, J. S., Algebraic and Arithmetic Groups.
Mumford, David (1970), Abelian varieties, Oxford University Press, ISBN 978-0-19-560528-0, OCLC 138290
Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR1642713
Waterhouse, William C. (1979), Introduction to affine group schemes, Graduate Texts in Mathematics, 66, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90421-4
Weil, André (1971), Courbes algébriques et variétés abéliennes, Paris: Hermann, OCLC 322901