Questions tagged [algebraic-groups]

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.