# Diagonalizable algebraic group

An affine algebraic group $G$ that is isomorphic to a closed subgroup of an algebraic torus. Thus, $G$ is isomorphic to a closed subgroup of a multiplicative group of all diagonal matrices of given size. If $G$ is defined over a field $k$ and the isomorphism is defined over $k$ , the diagonalizable algebraic group $G$ is said to be split (or decomposable) over $k$ .

Any closed subgroup $H$ in a diagonalizable algebraic group $G$ , as well as the image of $G$ under an arbitrary rational homomorphism $\phi$ , is a diagonalizable algebraic group. If, in addition, $G$ is defined and split over a field $k$ , while $\phi$ is defined over $k$ , then both $H$ and $\phi (G)$ are defined and split over $k$ .

A diagonalizable algebraic group is split over $k$ if and only if elements in the group $\widehat{G}$ of its rational characters are rational over $k$ . If $\widehat{G}$ contains no non-unit elements rational over $k$ , the diagonalizable algebraic group $G$ is said to be anisotropic over $k$ . Any diagonalizable algebraic group $G$ defined over the field $k$ is split over some finite separable extension of $k$ .

A diagonalizable algebraic group is connected if and only if it is an algebraic torus. The connectedness of $G$ is also equivalent to the absence of torsion in $\widehat{G}$ . For any diagonalizable algebraic group $G$ defined over $k$ , the group $\widehat{G}$ is a finitely-generated Abelian group without $p$ - torsion, where $p$ is the characteristic of $k$ .

Any diagonalizable algebraic group $G$ which is defined and split over a field $k$ is the direct product of a finite Abelian group and an algebraic torus defined and split over $k$ . Any diagonalizable algebraic group $G$ which is connected and defined over a field $k$ contains a largest anisotropic subtorus $G _{a}$ and a largest subtorus $G _{d}$ which is split over $k$ ; for these, $G = G _{a} G _{d}$ , and $G _{a} \cap G _{d}$ is a finite set.

If a diagonalizable algebraic group $G$ is defined over a field $k$ and $\Gamma$ is the Galois group of the separable closure of $k$ , then $\widehat{G}$ is endowed with a continuous action of $\Gamma$ . If, in addition, $\phi : \ G \rightarrow H$ is a rational homomorphism between diagonalizable algebraic groups, while $G$ , $H$ and $\phi$ are defined over $k$ , then the homomorphism $\widehat \phi : \ \widehat{H} \rightarrow \widehat{G}$ is $\Gamma$ - equivariant (i.e. is a homomorphism of $\Gamma$ - modules). The resulting contravariant functor from the category of diagonalizable $\Gamma$ - groups and their $k$ - morphisms into the category of finitely-generated Abelian groups without $k$ - torsion with a continuous action of the group $p$ and their $\Gamma$ - equivariant homomorphisms is an equivalence of these categories.

#### References

 [1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 [2] T. Ono, "Arithmetic of algebraic tori" Ann. of Math. , 74 : 1 (1961) pp. 101–139 MR0124326 Zbl 0119.27801