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.

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