User:Maximilian Janisch/latexlist/Algebraic Groups/Diagonalizable algebraic group

This page is a copy of the article Diagonalizable algebraic group in order to test automatic LaTeXification. This article is not my work.

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

Any closed subgroup $H$ in a diagonalizable algebraic group $k$, as well as the image of $k$ under an arbitrary rational homomorphism $( 1 )$, is a diagonalizable algebraic group. If, in addition, $k$ is defined and split over a field $k$, while $( 1 )$ 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 $\vec { C }$ of its rational characters are rational over $k$. If $\vec { C }$ contains no non-unit elements rational over $k$, the diagonalizable algebraic group $k$ is said to be anisotropic over $k$. Any diagonalizable algebraic group $k$ 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 $k$ is also equivalent to the absence of torsion in $\vec { C }$. For any diagonalizable algebraic group $k$ defined over $k$, the group $\vec { C }$ is a finitely-generated Abelian group without $D$-torsion, where $D$ is the characteristic of $k$.

Any diagonalizable algebraic group $k$ 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 $k$ 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, $G = G _ { \mathscr { L } } G _ { \mathscr { G } }$, and $G _ { a k } \cap G _ { Q }$ is a finite set.

If a diagonalizable algebraic group $k$ is defined over a field $k$ and $I$ is the Galois group of the separable closure of $k$, then $\vec { C }$ is endowed with a continuous action of $I$. If, in addition, $\phi : G \rightarrow H$ is a rational homomorphism between diagonalizable algebraic groups, while $k$, $H$ and $( 1 )$ are defined over $k$, then the homomorphism $\hat { \phi } : \hat { H } \rightarrow \hat { G }$ is $I$-equivariant (i.e. is a homomorphism of $I$-modules). The resulting contravariant functor from the category of diagonalizable $k$-groups and their $k$-morphisms into the category of finitely-generated Abelian groups without $D$-torsion with a continuous action of the group $I$ and their $I$-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

References

[a1]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039