Form of an algebraic group

2020 Mathematics Subject Classification: Primary: 14L [MSN][ZBL]

A form of an algebraic group $G$ defined over a field $k$ is an algebraic group $G'$ defined over $k$ and isomorphic to $G$ over some extension $L$ of $k$. In this case $G'$ is called an $L/k$-form of $G$. If $K_s$ is the separable closure of $k$ in a fixed algebraically closed ground field $K$ (a universal domain), then $k_s/k$-forms are simply called $k$-forms of $G$. Two $L/k$-forms of a group are said to be equivalent if they are isomorphic over $k$. The set of equivalence classes of $L/k$-forms of $G$ is denoted by $E(L/k,G)$ (in the case $L=k_s$ by $E(k,G)$) (see [Vo], [Ti], [Sp]).

Example. Let $k=\R$, $K=\C$. Then

$$G' = \Big\{ \begin{pmatrix} x & y\\ -y & x\end{pmatrix} : x^2+y^2 = 1 \Big\}$$ and

$$G = \{\def\diag{ {\rm diag}}\diag (x.y) : xy=1\}$$ are two subgroups of the general linear group $\def\GL{ {\rm GL}}\GL$ defined over $k$, and $G'$ is a $k$-form of $G$ (the isomorphism $\def\phi{\varphi} : G' \to G$, defined over $K$, is given by the formula

$$\phi \Big(\begin{pmatrix} x & y\\ -y & x\end{pmatrix}\Big) = \diag(x+iy,x-iy).$$ This $k$-form is not equivalent to $G$ (if one regards $G$ as a $k$-form of itself relative to the identity isomorphism $G\to G$). In this example, the set $E(k,G)$ consists of the two elements represented by the two $k$-forms above.

The problem of classifying forms of algebraic groups can be naturally reformulated in the language of Galois cohomology, [Se], [Vo]. Namely, suppose that $L/k$ is a Galois extension with Galois group $\def\G{\Gamma}\G_{L/k}$ (equipped with the Krull topology). The group $\G_{L/k}$ acts naturally on the group $\def\Aut{ {\rm Aut}}\Aut_L(G)$ of all $L$-automorphisms of $G$, and also on the set of all $L$-isomorphisms from $G'$ to $G$ (in coordinates, these actions reduce to applying the automorphisms in $\G_{L/k}$ to the coefficients of the rational functions defining the respective mappings). Let $\phi: G'\to G$ be some $L$-isomorphism, let $\def\s{\sigma}\s\in\G_{L/k}$ and let $\phi^\s$ be the image of $\phi$ under the action of $\s$. Then the mapping $\G_{L/k}\to\Aut_L G$, $\s \mapsto c_\s= \phi^\s\circ \phi^{-1}$, is a continuous $1$-cocycle of $\G_{L/k}$ with values in the discrete group $\Aut_L G$. When replacing $\phi$ by another $L$-isomorphism $G'\to G$, this cocycle changes to a cocycle in the same cohomology class. Thus arises a mapping $E(L/k, G)\to {\rm H}^1(\G_{L/k},\Aut_L G)$. The main importance of the cohomological interpretation of the forms of $G$ consists in the fact that this mapping is bijective. In the case when all automorphisms $c_\s$ are inner, $G'$ is called an inner form of $G$, 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: $k$-roots, the $k$-Weyl group, the Bruhat decomposition over $k$, etc. Here the role of maximal [algebraic torus|tori]] is played by maximal $k$-split tori, and that of Borel subgroups by minimal $k$-parabolic subgroups [Bo], [Hu], , [Ti]. This theory enables one to reduce the question of classifying forms to that of classifying anisotropic reductive groups over $k$ (see Anisotropic group; Anisotropic kernel). The question of classifying the latter depends essentially on the properties of the field $k$. If $k=\R$ and $K=\C$, 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

[Bo]	A. Borel, "Linear algebraic groups", Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[BoTi]	A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES, 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[BoTi2]	A. Borel, J. Tits, "Complement à l'article "Groupes réductifs" " Publ. Math. IHES, 41 (1972) pp. 253–276 MR0315007
[DeGa]	M. Demazure, P. Gabriel, "Groupes algébriques", 1, Masson (1970) MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401
[Hu]	J.E. Humphreys, "Linear algebraic groups", Springer (1975) MR0396773 Zbl 0325.20039
[Ja]	J.C. Jantzen, "Representations of algebraic groups", Acad. Press (1987) MR0899071 Zbl 0654.20039
[Se]	J.-P. Serre, "Cohomologie Galoisienne", Springer (1964) MR0180551 Zbl 0128.26303
[Se2]	J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959) MR0103191
[Se3]	J.-P. Serre, "Local fields", Springer (1979) (Translated from French) MR0554237 Zbl 0423.12016
[Sp]	T.A. Springer, "Reductive groups", Proc. Symp. Pure Math., 33 : 1, Amer. Math. Soc. (1979) pp. 3–27 MR0546587 Zbl 0416.20034
[Ti]	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
[Vo]	V.E. Voskresenskii, "Algebraic tori", Moscow (1977) (In Russian) MR0506279 MR0472845 Zbl 0379.14001 Zbl 0367.14007