View source for Algebraic group

A group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a0115101.png" /> provided with the structure of an [[Algebraic variety|algebraic variety]] in which the multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a0115102.png" /> and the inversion mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a0115103.png" /> are regular mappings (morphisms) of algebraic varieties. An algebraic group is said to be defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a0115104.png" /> if its underlying algebraic variety and the morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a0115105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a0115106.png" /> are defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a0115107.png" />. In such a case the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a0115108.png" />-rational points of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a0115109.png" /> is an (abstract) group which is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151010.png" />. An algebraic group is called connected if its algebraic variety is connected. The dimension of an algebraic group is the dimension of its algebraic variety. In what follows, only connected algebraic groups will be considered. A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151011.png" /> of an algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151012.png" /> is called algebraic if it is a closed subvariety of the algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151013.png" />. For such subgroups the space of cosets (left or right) can be naturally provided with the structure of an algebraic variety, defined by a universal property (cf. [[Quotient space|Quotient space]] of an algebraic group). If the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151014.png" /> is also normal, then the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151015.png" /> is an algebraic group with respect to this structure, and is called an algebraic quotient group. A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151016.png" /> of algebraic groups is called algebraic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151017.png" /> is a morphism of their algebraic varieties; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151018.png" /> is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151019.png" />, it is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151021.png" />-homomorphism. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151023.png" />-isomorphism of an algebraic group is defined in a similar manner.

Examples of algebraic groups: The general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151024.png" /> (the group of all invertible matrices of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151025.png" />, with coefficients in a fixed algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151026.png" />); the group of triangular matrices; an [[Elliptic curve|elliptic curve]].

There are two main types of algebraic groups, with altogether different properties: Abelian varieties and linear algebraic groups (cf. [[Abelian variety|Abelian variety]]; [[Linear algebraic group|Linear algebraic group]]). The type of a particular group is determined exclusively by properties of its variety. An algebraic group is called an Abelian variety if its algebraic variety is complete. An algebraic group is called linear if it is isomorphic to an algebraic subgroup of a general linear group. An algebraic group is linear if and only if its algebraic variety is affine. These two classes of algebraic groups have a trivial intersection: If an algebraic group is both an Abelian variety and a linear group, then it is the identity group. The study of arbitrary algebraic groups reduces to a great extent to the study of Abelian varieties and linear groups. In particular, an arbitrary algebraic group contains a unique normal linear algebraic subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151027.png" /> such that the quotient group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011510/a01151028.png" /> is an Abelian variety [[#References|[a1]]]. Numerous examples of algebraic groups which are neither linear algebraic groups nor Abelian varieties are given by the theory of generalized Jacobi varieties for algebraic curves with singular points [[#References|[3]]] (cf. [[Jacobi variety|Jacobi variety]]). A natural extension of the class of algebraic groups leads to the concept of a [[Group scheme|group scheme]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,   "Linear algebraic groups" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Mumford,   "Abelian varieties" , Oxford Univ. Press  (1974)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-P. Serre,   "Groupes algébrique et corps des classes" , Hermann  (1959)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  C. Chevalley,   "La théorie des groupes algébriques"  J.A. Todd (ed.) , ''Proc. Internat. Congress Mathematicians (Edinburgh, 1958)'' , Cambridge Univ. Press  (1960)  pp. 53–68</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M. Demazure,   A. Grothendieck,   "Schémas en groupes" , ''Sem. Geom. Alg. 1963–1964'' , ''Lect. notes in math.'' , '''151–153''' , Springer  (1970)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Chevalley,   "Une démonstration d'un théorème sur les groupes algébriques"  ''J. Math. Pures Appl.'' , '''39'''  (1960)  pp. 307–317</TD></TR></table>