Form of an algebraic group
defined over a field
An algebraic group defined over
and isomorphic to
over some extension
of
. In this case
is called an
-form of
. If
is the separable closure of
in a fixed algebraically closed ground field
(a universal domain), then
-forms are simply called
-forms of
. Two
-forms of a group are said to be equivalent if they are isomorphic over
. The set of equivalence classes of
-forms of
is denoted by
(in the case
by
) (see [5], [7], [8]).
Example. Let ,
. Then
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and
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are two subgroups of the general linear group defined over
, and
is a
-form of
(the isomorphism
, defined over
, is given by the formula
![]() |
This -form is not equivalent to
(if one regards
as a
-form of itself relative to the identity isomorphism
). In this example, the set
consists of the two elements represented by the two
-forms above.
The problem of classifying forms of algebraic groups can be naturally reformulated in the language of Galois cohomology, [3], [5]. Namely, suppose that is a Galois extension with Galois group
(equipped with the Krull topology). The group
acts naturally on the group
of all
-automorphisms of
, and also on the set of all
-isomorphisms from
to
(in coordinates, these actions reduce to applying the automorphisms in
to the coefficients of the rational functions defining the respective mappings). Let
be some
-isomorphism, let
and let
be the image of
under the action of
. Then the mapping
,
, is a continuous
-cocycle of
with values in the discrete group
. When replacing
by another
-isomorphism
, this cocycle changes to a cocycle in the same cohomology class. Thus arises a mapping
. The main importance of the cohomological interpretation of the forms of
consists in the fact that this mapping is bijective. In the case when all automorphisms
are inner,
is called an inner form of
, and otherwise an outer form.
For connected reductive groups there is a thoroughly developed theory of forms, where relative versions of the structure theory of linear algebraic groups over an algebraically closed field are established: -roots, the
-Weyl group, the Bruhat decomposition over
, etc. Here the role of maximal tori is played by maximal
-split tori, and that of Borel subgroups by minimal
-parabolic subgroups [1], [2], , [7]. This theory enables one to reduce the question of classifying forms to that of classifying anisotropic reductive groups over
(see Anisotropic group; Anisotropic kernel). The question of classifying the latter depends essentially on the properties of the field
. If
and
, then the characterization of forms of semi-simple algebraic groups is the same as that of real forms of complex semi-simple algebraic groups (see Complexification of a Lie group).
References
[1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) |
[2] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) |
[3] | J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) |
[4] | J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) |
[5] | V.E. Voskresenskii, "Algebraic tori" , Moscow (1977) (In Russian) |
[6a] | A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 |
[6b] | A. Borel, J. Tits, "Complement à l'article "Groupes réductifs" " Publ. Math. IHES , 41 (1972) pp. 253–276 |
[7] | J. Tits, "Classification of algebraic semi-simple groups" , Algebraic Groups and Discontinuous Subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) pp. 33–62 |
[8] | T.A. Springer, "Reductive groups" , Proc. Symp. Pure Math. , 33 : 1 , Amer. Math. Soc. (1979) pp. 3–27 |
[9] | J.-P. Serre, "Local fields" , Springer (1979) (Translated from French) |
Comments
References
[a1] | M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970) |
[a2] | J.C. Jantzen, "Representations of algebraic groups" , Acad. Press (1987) |
Form of an algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Form_of_an_algebraic_group&oldid=17664