# Algebraic group

A group $G$ provided with the structure of an algebraic variety in which the multiplication $\mu: G\times G \to G$ and the inversion mapping $\nu: G \to G$ are regular mappings (morphisms) of algebraic varieties. An algebraic group is said to be defined over a field $k$ if its underlying algebraic variety and the morphisms $\mu$ and $\nu$ are defined over $k$. In such a case the set of $k$-rational points of the variety $G$ is an (abstract) group which is denoted by $G(k)$. 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 $H$ of an algebraic group $G$ is called algebraic if it is a closed subvariety of the algebraic variety $G$. 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 of an algebraic group). If the subgroup $H$ is also normal, then the quotient group $G/H$ is an algebraic group with respect to this structure, and is called an algebraic quotient group. A homomorphism $\phi: G\to \tilde G$ of algebraic groups is called algebraic if $\phi$ is a morphism of their algebraic varieties; if $\phi$ is defined over $k$, it is called a $k$-homomorphism. A $k$-isomorphism of an algebraic group is defined in a similar manner.
Examples of algebraic groups: The general linear group ${\rm GL}(n,k)$ (the group of all invertible matrices of order $n$, with coefficients in a fixed algebraically closed field $k$); the group of triangular matrices; an elliptic curve.
There are two main types of algebraic groups, with altogether different properties: Abelian varieties and linear algebraic groups (cf. Abelian variety; 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 a complete algebraic variety. 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 $H$ such that the quotient group $G/H$ is an Abelian variety [Ch2]. 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 [Se] (cf. Jacobi variety). A natural extension of the class of algebraic groups leads to the concept of a group scheme.