View source for Linear algebraic group


{{MSC|14|14L}}
{{TEX|done}}


A ''linear algebraic group'' is
an
[[Algebraic group|algebraic group]] that is isomorphic to an algebraic
subgroup of a
[[General linear group|general linear group]]. An algebraic group $G$
is linear if and only if the
[[Algebraic variety|algebraic variety]] $G$ is affine, that is,
isomorphic to a Zariski-closed subvariety of an affine space (cf. also
[[Zariski topology|Zariski topology]]).


The theory of linear algebraic groups arose in the context of the
Galois theory of solving linear differential equations by quadratures
at the end of 19th century (S. Lie, E. Picard, L. Maurer), and the
study of linear algebraic groups over the field of complex numbers was
originally carried out by analogy with the theory of Lie groups by the
method of Lie algebras. A linear algebraic group $G$ over the field of
complex numbers $\mathbb C$ can be regarded as an analytic subgroup (cf.
[[Analytic group|Analytic group]]) of a group ${\rm GL}(n,\mathbb C)$ (of invertible
complex $n\times n$-matrices). The
[[Lie algebra|Lie algebra]] of $G$, defined in the usual way, is then
a subalgebra of the Lie algebra of ${\rm GL}(n,\mathbb C)$. An exhaustive answer can be
given to the question arising here — which Lie subalgebras correspond
to algebraic (and not only analytic) subgroups of ${\rm GL}(n,\mathbb C)$ (see
[[Lie algebra, algebraic|Lie algebra, algebraic]] 1)).

In the middle of the 20th century the methods of the theory of Lie
groups and Lie algebras were completed and generalized by
C. Chevalley. He subsequently applied these methods to the study of
linear algebraic groups over arbitrary fields of characteristic zero
(see
{{Cite|Ch}}). For fields of non-zero characteristic the method
of Lie algebras is less effective, so there naturally arose the need
for a global investigation of linear algebraic groups by means of
methods of algebraic geometry. The foundations of a global
investigation of linear algebraic groups were laid by A. Borel (see
{{Cite|Bo2}}), after which the theory of linear algebraic
groups acquired the form of an orderly discipline (see
{{Cite|Ch3}}). One of the main problems in the theory of linear
algebraic groups is that of classifying linear algebraic groups up to
isomorphism. To some extent this problem reduces to the classification
of two types of linear algebraic groups, semi-simple and solvable,
since an arbitrary linear algebraic group has a maximal connected
solvable normal subgroup $H$ such that the quotient group $G/H$ is a
semi-simple linear algebraic group. One of the main results in the
theory of linear algebraic groups is Chevalley's classification of
connected semi-simple linear algebraic groups over an algebraically
closed field of arbitrary characteristic (see
{{Cite|Ch2}},
{{Cite|Hu}}). This classification is analogous to the
classification of Cartan–Killing of complex semi-simple Lie algebras
(cf.
[[Lie algebra, semi-simple|Lie algebra, semi-simple]]). The
classification of Chevalley is based on the fact that in a semi-simple
algebraic group one can construct analogues to the elements of the
theory of Cartan–Killing, namely the Cartan subgroups (cf.
[[Cartan subgroup|Cartan subgroup]]), roots, etc. An important role is
played here by Borel subgroups (cf.
[[Borel subgroup|Borel subgroup]]) and maximal tori (cf.
[[Algebraic torus|Algebraic torus]]).

Let $G$ be a semi-simple linear algebraic group (cf.
[[Semi-simple algebraic group|Semi-simple algebraic group]]), $T$ a
maximal torus, $N_G(T)$ the normalizer of $T$ in $G$ (cf.
[[Normalizer of a subset|Normalizer of a subset]]), and let $N_G(T)/T$ be the
[[Weyl group|Weyl group]] of $G$. The torus $T$ is contained in only
finitely many Borel subgroups of $G$, which are permuted transitively
under conjugation by elements of $N_G(T)$. One can construct certain
monomorphisms of the additive group of the field into the Borel
subgroups (containing $T$), which play the role of roots. The
[[Bruhat decomposition|Bruhat decomposition]] of $G$ gives insight
into the purely group-theoretical aspects of semi-simple groups (see
{{Cite|Bo}},
{{Cite|Ch2}},
{{Cite|Hu}}). The final classification of semi-simple groups
does not depend on the characteristic of the ground field and
therefore coincides with the classification of complex semi-simple
algebraic groups.

The classification up to $k $-isomorphism of semi-simple linear
algebraic groups defined over a non-algebraically closed field $k$ is
significantly more complicated; it depends essentially on the field
$k$ and reduces to the classification of $k$-forms of algebraic groups
(see
[[Form of an algebraic group|Form of an algebraic group]]). The set of
$k$-forms of an algebraic group $G$ defined over an algebraic closure
$K$ of $k$ is in one-to-one correspondence with the one-dimensional
[[Galois cohomology|Galois cohomology]] set $H^1(k,{\rm Aut}(G))$, where ${\rm Aut}(G)$ is the
group of automorphisms of $G$. In those cases when the
[[Galois group|Galois group]] ${\rm Gal}(K/k)$ of the extension $K/k$ is known (for
example, if $k$ is the field of real numbers or a finite field), this
gives significant information about the forms of $G$. But the main
difficulty of investigating this deep problem, which is still (1989)
far from being solved, lies in the study of the $k$-structures of
semi-simple linear algebraic groups.

In the case of an arbitrary field $k$, the maximal $k$-split tori play
a role analogous to that of maximal algebraic tori in a group $G$ over
an algebraically closed field. A $k$-split torus is a torus that is
$k$-isomorphic to a direct product of one-dimensional groups ${\rm GL(1,k)}$ which
are defined over $k$. Maximal $k$-split tori in $G$ are conjugate by
elements of the group of $k$-rational points $G_k$ and their dimension
is called the $k$-rank of $G$ (denoted by ${\rm rank}_k\;G$). If ${\rm rank}_k\;G > 0$, then $G$ is
said to be $k$-isotropic, otherwise $k$-anisotropic. A semi-simple
group $G$ is $k$-isotropic if and only if $G_k$ has unipotent elements
other than the identity. For a simple simply-connected group $G$ there
was the
[[Kneser–Tits hypothesis|Kneser–Tits hypothesis]] that if ${\rm rank}_k\;G > 0$, then
$G_k$ is generated by unipotent elements (see
{{Cite|Ti2}}). In general it has been proved false (see
{{Cite|Pl3}}). The role of a Borel subgroup in the case of an
arbitrary field $k$ is played by a minimal parabolic $k$-subgroup,
that is, a subgroup of $G$ containing a Borel subgroup which is
defined over $k$ and is minimal for these properties. One can define
the root system with respect to a maximal $k$-split torus in $G$ and a
relative Weyl group (see
{{Cite|Bo3}}). If $G$ has a $k$-split maximal torus, then these
structural elements do not depend on the field $k$ and also determine
the group up to $k$-isomorphism. Groups that have $k$-split maximal
tori are said to be $k$-split or Chevalley groups (cf.
[[Chevalley group|Chevalley group]]). Chevalley's classification of
semi-simple groups over an algebraically closed field carries over to
$k$-split groups. Chevalley proved that every semi-simple group has a
split $k$-form (even "a form over Z" , see
{{Cite|Ch2}},
{{Cite|St}}).

In the general case Borel and J. Tits
{{Cite|Bo3}} proved the existence of an analogue of the Bruhat
decomposition for a group $G_k$, in which the role of the Borel
subgroups is played by the groups of $k$-rational points of minimal
parabolic $k$-subgroups. This makes it possible to reduce to a large
extent the classification of semi-simple linear algebraic groups of
positive $k$-rank to the classification of semi-simple groups of
$k$-rank zero, that is, to the classification of $k$-anisotropic
groups. Namely, a semi-simple group $G$ is defined up to
$k$-isomorphism by its isomorphism class over the algebraic closure of
$k$, its $k$-index and the semi-simple $k$-anisotropic kernel (see
[[Anisotropic kernel|Anisotropic kernel]]) (see
{{Cite|Ti}}). In some cases it has been possible to obtain a
classification of $k$-anisotropic semi-simple groups. For example,
over a finite field there are no anisotropic semi-simple groups, and
for a wide class of local fields $k$ it has been proved (see ) that
every $k$-anisotropic simple linear algebraic group $G$ is an inner
form of type ${\rm A}_n$, that is, in the simply-connected case $G_k$ is
isomorphic to ${\rm SL}(1,D)$ where $D$ is a skew-field of finite degree over
$k$. The basis of these results is the concept of a
[[Tits system|Tits system]], which is a deep axiomatic generalization
of the Bruhat decomposition in the classical case (see
{{Cite|Ti2}}).

Among other general results one should mention Grothendieck's results
that a linear algebraic group defined over $k$ has maximal tori that
are defined over $k$, and that a connected reductive linear algebraic
group over $k$ is unirational over $k$, as an algebraic variety (see
{{Cite|Bo}},
{{Cite|DeGr}},
{{Cite|Pl2}}).

If $G$ is defined over an algebraic number field or over a field of
algebraic functions in one variable, then there arise problems about
the arithmetic properties of $G$, the study of which is the subject of
the arithmetic theory of linear algebraic groups (see
[[Linear algebraic groups, arithmetic theory of|Linear algebraic groups, arithmetic theory of]],
{{Cite|Pl2}}).

The ideas and techniques of linear algebraic groups have been used to
study arbitrary linear groups, which has led to one of the fundamental
methods in the theory of linear groups (see
{{Cite|Pl}}).

====References====
{|
|-
|valign="top"|{{Ref|Bo}}||valign="top"| A. Borel, "Linear algebraic groups", Benjamin (1969)  {{MR|0251042}}  {{ZBL|0206.49801}} {{ZBL|0186.33201}}                
|-
|valign="top"|{{Ref|Bo2}}||valign="top"| A. Borel, "Groupes linéaires algébriques" ''Ann. of Math. (2)'', '''64''' (1956) pp. 20–82 {{MR|0093006}} 
|-
|valign="top"|{{Ref|Bo3}}||valign="top"| A. Borel, "Groupes réductifs" ''Publ. Math. IHES'', '''27''' (1965) pp. 55–150  {{MR|0207712}}  {{ZBL|0145.17402}}                
|-
|valign="top"|{{Ref|BoSp}}||valign="top"| A. Borel, T.A. Springer, "Rationality properties of linear algebraic groups II" ''Tohoku Math. J.'', '''20''' (1968) pp. 443–497  {{MR|0244259}}  {{ZBL|0211.53302}}                
|-
|valign="top"|{{Ref|BrTi}}||valign="top"| F. Bruhat, J. Tits, "BN paires de type affine et données radicielles" ''C.R. Acad. Sci. Paris'', '''263''' (1966) pp. 598–601    {{MR|0242833}}  {{ZBL|0214.28803}}             
|-
|valign="top"|{{Ref|BrTi2}}||valign="top"| F. Bruhat, J. Tits, "Groupes simples résiduellement déployés sur un corps local" ''C.R. Acad. Sci. Paris'', '''263''' (1966) pp. 766–768  {{MR|0242834}}  {{ZBL|0263.14013}}                
|-
|valign="top"|{{Ref|BrTi3}}||valign="top"| F. Bruhat, J. Tits, "Groupes algébriques simples sur un corps local" ''C.R. Acad. Sci. Paris'', '''263''' (1966) pp. 822–825           {{MR|0242836}} {{MR|0242835}}  {{ZBL|0263.14015}} {{ZBL|0263.1x4014}}       
|-
|valign="top"|{{Ref|BrTi4}}||valign="top"| F. Bruhat, J. Tits, "Groupes algébriques simples sur un corps local: Cohomologie galoisienne, décompositions d'Iwasawa et de Cartan" ''C.R. Acad. Sci. Paris'', '''263''' (1966) pp. 867–869           {{MR|0242836}}  {{ZBL|0263.14015}}       
|-
|valign="top"|{{Ref|BrTi5}}||valign="top"| F. Bruhat, J. Tits, "Groupes réductifs sur un corps local. I" ''Publ. Math.IHES'', '''41''' (1972) pp. 5–251              {{MR|0327923}}     
|-
|valign="top"|{{Ref|BrTi6}}||valign="top"| F. Bruhat, J. Tits, "Groupes réductifs sur un corps local. II" ''Publ. Math.IHES'', '''60''' (1984) pp. 5–184              {{MR|0756316}}  {{ZBL|0597.14041}}    
|-
|valign="top"|{{Ref|BrTi7}}||valign="top"| F. Bruhat, J. Tits, "Groupes algébriques sur un corps local. III" ''J. Fac. Sci. Tokyo Univ.'', '''34''' (1987) pp. 671–698              {{MR|0927605}}  {{ZBL|0657.20040}}    
|-
|valign="top"|{{Ref|Ch}}||valign="top"| C. Chevalley, "Théorie des groupes de Lie", '''2''', Hermann (1951)  {{MR|0051242}}  {{ZBL|0054.01303}}                
|-
|valign="top"|{{Ref|Ch3}}||valign="top"| C. Chevalley, "La théorie des groupes algébriques" J.A. Todd (ed.), ''Proc. Internat. Congress Mathematicians (Edinburgh, 1958)'', Cambridge Univ. Press (1960) pp. 53–68                  
|-
|valign="top"|{{Ref|DeGr}}||valign="top"| M. Demazure, A. Grothendieck, "Schémas en groupes. I-III", ''Lect. notes in math.'', '''151–153''', Springer (1970)        {{MR|0274458}} {{MR|0274459}} {{MR|0274460}}           
|-
|valign="top"|{{Ref|Hu}}||valign="top"| J.E. Humphreys, "Linear algebraic groups", Springer (1981)   {{ZBL|0471.20029}}                
|-
|valign="top"|{{Ref|Pl}}||valign="top"| V.P. Platonov, "The theory of algebraic linear groups and periodic groups" ''Transl. Amer. Math. Soc. (2)'', '''69''' (1968) pp. 61–110 ''Izv. Akad. Nauk SSSR Ser. Mat.'', '''30''' (1966) pp. 573–620                  
|-
|valign="top"|{{Ref|Pl2}}||valign="top"| V.P. Platonov, "Algebraic groups" ''J. Soviet Math.'', '''4''' : 5 (1975) pp. 463–482 ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'', '''11''' (1974) pp. 5–36  {{MR|0466334}}  {{ZBL|0386.20019}} {{ZBL|0305.20023}}                
|-
|valign="top"|{{Ref|Pl3}}||valign="top"| V.P. Platonov, "The Tannaka–Artin problem and reduced $K$-theory" ''Math. USSR Izv.'', '''10''' (1976) pp. 211–243 ''Izv. Akad. Nauk SSSR Ser. Mat.'', '''40''' : 2 (1976) pp. 227–261  {{MR|0424872}} {{MR|0412228}} {{MR|0407082}}                 
|-
|valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959)  {{MR|0103191}}                 
|-
|valign="top"|{{Ref|Se2}}||valign="top"| J.-P. Serre, "Cohomologie Galoisienne", Springer (1964)  {{MR|0181643}}  {{ZBL|0143.05901}} {{ZBL|0128.26303}}                
|-
|valign="top"|{{Ref|St}}||valign="top"| R.G. Steinberg, "Lectures on Chevalley groups", Yale Univ. Press (1968)  {{MR|0466335}}  {{ZBL|1196.22001}}                
|-
|valign="top"|{{Ref|Ti}}||valign="top"| J. Tits, "Classification of algebraic simple groups", ''Algebraic Groups and Discontinuous Subgroups'', ''Proc. Symp. Pure Math.'', '''9''', Amer. Math. Soc.  (1966) pp. 33–62                  
|-
|valign="top"|{{Ref|Ti2}}||valign="top"| J. Tits, "Algebraic and abstract simple groups" ''Ann. of Math. (2)'', '''80''' (1964) pp. 313–329  {{MR|0093006}}
|-
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