Diagonalizable algebraic group
An affine algebraic group that is isomorphic to a closed subgroup of an algebraic torus. Thus,
is isomorphic to a closed subgroup of a multiplicative group of all diagonal matrices of given size. If
is defined over a field
and the isomorphism is defined over
, the diagonalizable algebraic group
is said to be split (or decomposable) over
.
Any closed subgroup in a diagonalizable algebraic group
, as well as the image of
under an arbitrary rational homomorphism
, is a diagonalizable algebraic group. If, in addition,
is defined and split over a field
, while
is defined over
, then both
and
are defined and split over
.
A diagonalizable algebraic group is split over if and only if elements in the group
of its rational characters are rational over
. If
contains no non-unit elements rational over
, the diagonalizable algebraic group
is said to be anisotropic over
. Any diagonalizable algebraic group
defined over the field
is split over some finite separable extension of
.
A diagonalizable algebraic group is connected if and only if it is an algebraic torus. The connectedness of is also equivalent to the absence of torsion in
. For any diagonalizable algebraic group
defined over
, the group
is a finitely-generated Abelian group without
-torsion, where
is the characteristic of
.
Any diagonalizable algebraic group which is defined and split over a field
is the direct product of a finite Abelian group and an algebraic torus defined and split over
. Any diagonalizable algebraic group
which is connected and defined over a field
contains a largest anisotropic subtorus
and a largest subtorus
which is split over
; for these,
, and
is a finite set.
If a diagonalizable algebraic group is defined over a field
and
is the Galois group of the separable closure of
, then
is endowed with a continuous action of
. If, in addition,
is a rational homomorphism between diagonalizable algebraic groups, while
,
and
are defined over
, then the homomorphism
is
-equivariant (i.e. is a homomorphism of
-modules). The resulting contravariant functor from the category of diagonalizable
-groups and their
-morphisms into the category of finitely-generated Abelian groups without
-torsion with a continuous action of the group
and their
-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 |
Comments
References
[a1] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039 |
Diagonalizable algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Diagonalizable_algebraic_group&oldid=21836