Algebraic torus

An algebraic torus of dimension $n$ is an algebraic group isomorphic over the base field to $({\mathbb{G}}_m{)}^n$. Its character and cocharacter lattices are dual free abelian groups $M$ and $N$.

A maximal algebraic torus is an algebraic torus maximal among the tori contained in an algebraic group. In a connected reductive algebraic group, all maximal algebraic tori are conjugate. A Borel subgroup contains maximal algebraic tori, and its unipotent radical gives a decomposition $B=T⋉U$.

A maximal algebraic torus $T$ is rational for $F$ when $F(T)=T$. Rationality does not require every point of $T$ to be fixed: the finite group $T^F$ is its rational-point group. Similarly a rational Borel subgroup or parabolic subgroup means an $F$-stable algebraic subgroup.

A rational maximal torus is elliptic when it is contained in no proper rational parabolic subgroup. This definition allows the split central torus of the ambient group; an absolute absence of every split subtorus is not the right condition when the centre is nontrivial.

The character lattice $M=\mathrm{Hom}(T,{\mathbb{G}}_m)$ of an algebraic torus is the free abelian group of algebraic group homomorphisms from $T$ to the multiplicative group.

An algebraic torus character is a morphism of algebraic groups $T\to {\mathbb{G}}_m$. For a split torus with character lattice $M$, these are precisely the Laurent monomials ${\chi }^m$, with ${\chi }^{m+m^{\prime }}={\chi }^m{\chi }^{m^{\prime }}$.

// Target: algebra.bigb

The cocharacter lattice $N=\mathrm{Hom}({\mathbb{G}}_m,T)$ is dual to the character lattice of an algebraic torus under composition: