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

and

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) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2]	J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039
[3]	J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0180551 Zbl 0128.26303
[4]	J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) MR0103191
[5]	V.E. Voskresenskii, "Algebraic tori" , Moscow (1977) (In Russian) MR0506279 MR0472845 Zbl 0379.14001 Zbl 0367.14007
[6a]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[6b]	A. Borel, J. Tits, "Complement à l'article "Groupes réductifs" " Publ. Math. IHES , 41 (1972) pp. 253–276 MR0315007
[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 MR224710
[8]	T.A. Springer, "Reductive groups" , Proc. Symp. Pure Math. , 33 : 1 , Amer. Math. Soc. (1979) pp. 3–27 MR0546587 Zbl 0416.20034
[9]	J.-P. Serre, "Local fields" , Springer (1979) (Translated from French) MR0554237 Zbl 0423.12016

References

[a1]	M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , Masson (1970) MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401
[a2]	J.C. Jantzen, "Representations of algebraic groups" , Acad. Press (1987) MR0899071 Zbl 0654.20039