Borel subgroups, maximal tori and Borel pairs

Let $k$ be a field and let $G$ be an affine algebraic group over $k$, that is, an affine group scheme of finite type over $k$ (Affine schemes and their coordinate rings, Group schemes of finite type over a field).

A torus of $G$ is a closed subgroup scheme $T\subseteq G$ (Morphisms and closed subgroup schemes of group schemes) whose base extension to a separable closure of $k$ is isomorphic to ${\mathbf{G}}_m^r$ for some integer $r\ge 0$; it is a split torus if that isomorphism is already defined over $k$ (Groups of multiplicative type and tori). It is a maximal torus if it is maximal with respect to inclusion among the tori of $G$. Equivalently, $T$ is a closed subgroup that is geometrically a product of copies of ${\mathbf{G}}_m$, and no strictly larger torus of $G$ contains it; for smooth connected affine groups, maximality is preserved by every field extension (Conrad, Grothendieck’s theorem on tori, Corollary 1.3, printed p. 1).

A Borel subgroup of a smooth $G$ is a smooth connected solvable closed subgroup scheme $B\subseteq G$ whose base extension to an algebraic closure is maximal among smooth connected solvable closed subgroup schemes. Over an algebraically closed field this means exactly that $B$ is a maximal connected solvable subgroup variety, with its reduced smooth structure (The derived subgroup, the derived series and solvable algebraic groups, Smooth morphism of schemes). Maximality here is among subgroup varieties, not arbitrary possibly infinitesimal subgroup schemes. A Borel subgroup need not be defined over a general field; existence and conjugacy are asserted below over an algebraically closed field.

A Borel pair is a pair $(B,T)$ consisting of a Borel subgroup $B$ and a maximal torus $T$ with $T\subseteq B$.

On this page Borel subgroups are used only for smooth $G$ (Smooth morphism of schemes); Borel subgroups are smooth by the subgroup-variety convention, and the existence and conjugacy theorems are proved later on this page. Maximal tori exist whenever $G$ has a torus, by maximizing dimension among tori, since a strict inclusion of tori increases dimension; the existence of a Borel subgroup containing a given torus is proved where it is used.