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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f0407901.png" /> defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f0407902.png" />''
+
{{MSC|14L}}
 +
{{TEX|done}}
  
An [[Algebraic group|algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f0407903.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f0407904.png" /> and isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f0407905.png" /> over some extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f0407906.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f0407907.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f0407908.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079010.png" />-form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079012.png" /> is the separable closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079013.png" /> in a fixed algebraically closed ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079014.png" /> (a universal domain), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079015.png" />-forms are simply called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079017.png" />-forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079018.png" />. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079019.png" />-forms of a group are said to be equivalent if they are isomorphic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079021.png" />. The set of equivalence classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079022.png" />-forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079023.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079024.png" /> (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079025.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079026.png" />) (see [[#References|[5]]], [[#References|[7]]], [[#References|[8]]]).
+
A ''form of an algebraic group
 +
$G$ defined over a field $k$'' is
 +
an
 +
[[Algebraic group|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 extension|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
 +
{{Cite|Vo}},
 +
{{Cite|Ti}},
 +
{{Cite|Sp}}).
  
Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079028.png" />. Then
+
Example. Let $k=\R$, $K=\C$. Then
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079029.png" /></td> </tr></table>
 
  
 +
$$G' = \Big\{ \begin{pmatrix} x & y\\ -y & x\end{pmatrix} : x^2+y^2 = 1 \Big\}$$
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079030.png" /></td> </tr></table>
+
$$G = \{\def\diag{ {\rm diag}}\diag (x.y) : xy=1\}$$
 
+
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
are two subgroups of the general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079031.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079032.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079033.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079034.png" />-form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079035.png" /> (the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079036.png" />, defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079037.png" />, is given by the formula
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079038.png" /></td> </tr></table>
 
 
 
This <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079039.png" />-form is not equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079040.png" /> (if one regards <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079041.png" /> as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079042.png" />-form of itself relative to the identity isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079043.png" />). In this example, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079044.png" /> consists of the two elements represented by the two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079045.png" />-forms above.
 
 
 
The problem of classifying forms of algebraic groups can be naturally reformulated in the language of [[Galois cohomology|Galois cohomology]], [[#References|[3]]], [[#References|[5]]]. Namely, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079046.png" /> is a Galois extension with Galois group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079047.png" /> (equipped with the Krull topology). The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079048.png" /> acts naturally on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079049.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079050.png" />-automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079051.png" />, and also on the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079052.png" />-isomorphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079053.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079054.png" /> (in coordinates, these actions reduce to applying the automorphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079055.png" /> to the coefficients of the rational functions defining the respective mappings). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079056.png" /> be some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079057.png" />-isomorphism, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079058.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079059.png" /> be the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079060.png" /> under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079061.png" />. Then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079063.png" />, is a continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079064.png" />-cocycle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079065.png" /> with values in the discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079066.png" />. When replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079067.png" /> by another <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079068.png" />-isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079069.png" />, this cocycle changes to a cocycle in the same cohomology class. Thus arises a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079070.png" />. The main importance of the cohomological interpretation of the forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079071.png" /> consists in the fact that this mapping is bijective. In the case when all automorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079072.png" /> are inner, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079073.png" /> is called an inner form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079074.png" />, 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: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079075.png" />-roots, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079076.png" />-Weyl group, the Bruhat decomposition over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079077.png" />, etc. Here the role of maximal tori is played by maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079078.png" />-split tori, and that of Borel subgroups by minimal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079079.png" />-parabolic subgroups [[#References|[1]]], [[#References|[2]]], , [[#References|[7]]]. This theory enables one to reduce the question of classifying forms to that of classifying anisotropic reductive groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079080.png" /> (see [[Anisotropic group|Anisotropic group]]; [[Anisotropic kernel|Anisotropic kernel]]). The question of classifying the latter depends essentially on the properties of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079081.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040790/f04079083.png" />, 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|Complexification of a Lie group]]).
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0180551}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) {{MR|0103191}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.E. Voskresenskii, "Algebraic tori" , Moscow (1977) (In Russian) {{MR|0506279}} {{MR|0472845}} {{ZBL|0379.14001}} {{ZBL|0367.14007}} </TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top"> A. Borel, J. Tits, "Complement à l'article "Groupes réductifs" " ''Publ. Math. IHES'' , '''41''' (1972) pp. 253–276 {{MR|0315007}} {{ZBL|}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> 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 {{MR|224710}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> T.A. Springer, "Reductive groups" , ''Proc. Symp. Pure Math.'' , '''33''' : 1 , Amer. Math. Soc. (1979) pp. 3–27 {{MR|0546587}} {{ZBL|0416.20034}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> J.-P. Serre, "Local fields" , Springer (1979) (Translated from French) {{MR|0554237}} {{ZBL|0423.12016}} </TD></TR></table>
 
 
 
  
 +
$$\phi \Big(\begin{pmatrix} x & y\\ -y & x\end{pmatrix}\Big) = \diag(x+iy,x-iy).$$
 +
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.
  
====Comments====
+
The problem of classifying forms of algebraic groups can be naturally reformulated in the language of
 +
[[Galois cohomology|Galois cohomology]],
 +
{{Cite|Se}},
 +
{{Cite|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 group|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: [[Root system|$k$-roots]], the [[Weyl group|$k$-Weyl group]], the [[Bruhat decomposition|Bruhat decomposition]] over $k$, etc. Here the role of maximal [[algebraic torus|tori]] is played by maximal $k$-split tori, and that of [[Borel subgroup|Borel subgroups]] by minimal [[Parabolic subgroup|$k$-parabolic subgroups]]
 +
{{Cite|Bo}},
 +
{{Cite|Hu}}, ,
 +
{{Cite|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 group]];
 +
[[Anisotropic kernel|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 group]]s is the same as that of real forms of complex semi-simple algebraic groups (see
 +
[[Complexification of a Lie group|Complexification of a Lie group]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , Masson (1970) {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.C. Jantzen, "Representations of algebraic groups" , Acad. Press (1987) {{MR|0899071}} {{ZBL|0654.20039}} </TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Bo}}||valign="top"| A. Borel, "Linear algebraic groups", Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}}
 +
|-
 +
|valign="top"|{{Ref|BoTi}}||valign="top"| A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'', '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}}
 +
|-
 +
|valign="top"|{{Ref|BoTi2}}||valign="top"| A. Borel, J. Tits, "Complement à l'article "Groupes réductifs" " ''Publ. Math. IHES'', '''41''' (1972) pp. 253–276 {{MR|0315007}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|DeGa}}||valign="top"| M. Demazure, P. Gabriel, "Groupes algébriques", '''1''', Masson (1970) {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}}
 +
|-
 +
|valign="top"|{{Ref|Hu}}||valign="top"| J.E. Humphreys, "Linear algebraic groups", Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}}
 +
|-
 +
|valign="top"|{{Ref|Ja}}||valign="top"| J.C. Jantzen, "Representations of algebraic groups", Acad. Press (1987) {{MR|0899071}} {{ZBL|0654.20039}}
 +
|-
 +
|valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre, "Cohomologie Galoisienne", Springer (1964) {{MR|0180551}} {{ZBL|0128.26303}}
 +
|-
 +
|valign="top"|{{Ref|Se2}}||valign="top"| J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959) {{MR|0103191}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|Se3}}||valign="top"| J.-P. Serre, "Local fields", Springer (1979) (Translated from French) {{MR|0554237}} {{ZBL|0423.12016}}
 +
|-
 +
|valign="top"|{{Ref|Sp}}||valign="top"| T.A. Springer, "Reductive groups", ''Proc. Symp. Pure Math.'', '''33''' : 1, Amer. Math. Soc. (1979) pp. 3–27 {{MR|0546587}} {{ZBL|0416.20034}}
 +
|-
 +
|valign="top"|{{Ref|Ti}}||valign="top"| 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 {{MR|224710}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|Vo}}||valign="top"| V.E. Voskresenskii, "Algebraic tori", Moscow (1977) (In Russian) {{MR|0506279}} {{MR|0472845}} {{ZBL|0379.14001}} {{ZBL|0367.14007}}
 +
|-
 +
|}

Latest revision as of 22:11, 26 December 2017

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 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

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[BoTi] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES, 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
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[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
How to Cite This Entry:
Form of an algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Form_of_an_algebraic_group&oldid=24149
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article