- Algebraic group
In

algebraic geometry , an**algebraic group**(or**group variety**) is a group that is analgebraic variety , such that the multiplication and inverse are given byregular function s on the variety. In category theoretic terms, an algebraic group is agroup object in the category of algebraic varieties.**Classes**Several important classes of groups are algebraic groups, including:

*Finite group s

* GL_{"n"}**C**, thegeneral linear group ofinvertible matrices over**C**

*Elliptic curve sTwo important classes of algebraic groups arise, that for the most part are studied separately: "abelian varieties" (the 'projective' theory) and "

linear algebraic group s" (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 theWeierstrass zeta function , or the theory ofgeneralized Jacobian s. But according to a basic theorem any algebraic group is an extension of anabelian variety by a linear algebraic group. This is a result ofClaude Chevalley : if "K" is aperfect 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. [*Chevalley's result is from 1960 and difficult. Contemporary treatment by*]Brian Conrad : [*http://www.math.lsa.umich.edu/~bdconrad/papers/chev.pdf PDF*] .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 thanLie group , when working over the field of real numbers: there are examples such as theuniversal 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 theidentity 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, agroup object in the category of schemes over "R". "Affine group scheme" is the concept dual to a type ofHopf 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 analgebraic group is a Zariski closedsubgroup .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 scheme s, in characteristic "p".**ee also***

Tame group

*Morley rank

*Cherlin-Zilber conjecture

*Adelic algebraic group **References*** | year=1972 | volume=21

*

* Milne, J. S., " [*http://www.jmilne.org/math/ Algebraic and Arithmetic Groups.*] "

*

* | year=1998 | volume=9

*

***Notes**

*Wikimedia Foundation.
2010.*