For questions about groups which have additional structure as algebraic varieties (the vanishing sets of collections of polynomials) which is compatible with their group structure. These include both linear algebraic groups as well as abelian varieties. Consider using with the (group-theory) tag.

This tag is for questions about algebraic groups. There are two main types of algebraic groups: linear algebraic groups and abelian varieties. The prototypical example of a linear algebraic group is $\mathrm{GL}_n$, the group of $n\times n$ matrices. The prototypical example of an abelian variety is an elliptic curve, which is the set of solutions to an equation $y^2 = x^3 + Ax + B$.

Over a field $k$, an algebraic group consists of (i) an underlying set $G$ defined as an algebraic subset of either affine space $G \subset \mathbb{A}^n_k$ (in the case of linear algebraic groups) or projective space $G \subset \mathbb{P}^n_k$ (in the case of abelian varieties) and (ii) a group operation called multiplication, which is a polynomial function $m\colon G \times G \to G$ satisfying axioms of associativity, invertibility, and identity. The set $G$ is referred to as an algebraic variety, and it is endowed with the Zariski topology, which is defined as the coarsest topology such that all subsets $Z$, which are cut out by the vanishing of a collection of polynomials, are closed. These closed subsets $Z \subset G$ are also algebraic varieties, called sub-varieties. If $Z$ is closed under the restriction of the multiplication map, i.e. if $m(Z \times Z) \subset Z$, then $Z$ also inherits a group structure and is called an algebraic subgroup of $G$.

Note that here we have not required a variety to be irreducible, which is a condition used by some authors. We have also not required $k$ to be algebraically closed, as some authors do.