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{{Group theory sidebar |Algebraic}}
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In [[mathematics]], a '''linear algebraic group'''  is a [[subgroup]] of the group of invertible ''n''&times;''n'' [[Matrix (mathematics)|matrices]] (under [[matrix multiplication]]) that is defined by [[polynomial]] equations. An example is the [[orthogonal group]], defined by the relation M<sup>T</sup>M = I where M<sup>T</sup> is the [[transpose]] of M.
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The main examples of linear algebraic groups are certain [[Lie group]]s, where the underlying [[Field (mathematics)|field]] is the [[Real number|real]] or [[Complex number|complex]] field. (For example, every compact Lie group can be regarded as the group of points of a real linear algebraic group, essentially by the [[Peter-Weyl theorem]].)
These were the first algebraic groups to be extensively studied. Such groups were known for a long time before their abstract algebraic theory was developed according to the needs of major applications. Compact Lie groups were considered by [[Élie Cartan]], [[Ludwig Maurer]], [[Wilhelm Killing]], and [[Sophus Lie]] in the 1880s and 1890s in the context of [[differential equation]]s and [[Galois theory]]. However, a purely algebraic theory was first developed by {{harvs|txt|last=Kolchin|author-link=Ellis Kolchin|year=1948}}, with [[Armand Borel]] as one of its pioneers. The [[Picard-Vessiot theory]] did lead to algebraic groups.
 
The first basic theorem of the subject is that any ''affine'' [[algebraic group]] is a linear algebraic group: that is, any [[affine variety]] V that has an algebraic group law has a ''faithful'' [[linear representation]], over the same field, which is also a morphism of varieties. For example the ''additive group'' of an ''n''-dimensional [[vector space]] has a faithful representation as (''n''+1)&times;(''n''+1) matrices.  
 
One can define the [[Lie algebra]] of an algebraic group purely algebraically (it consists of the [[dual number]] points based at the identity element); and this theorem shows that we get a matrix Lie algebra. A linear algebraic group G consists of a finite number of irreducible components, that are in fact also the connected components: the one G<sup>o</sup> containing the identity will be a [[normal subgroup]] of G.
 
One of the first uses for the theory was to define the [[Chevalley group]]s.
 
== Examples ==
Since <math>\mathbb{G}_m = GL_1</math>, <math>\mathbb{G}_m</math> is a linear algebraic group. The embedding <math>x \mapsto \begin{bmatrix}
1 & x \\
0 & 1
\end{bmatrix}</math>
shows that <math>\mathbb{G}_a</math> is a [[unipotent group]].
 
The deeper structure theory applies to connected linear algebraic groups G, and begins with the definition of [[Borel subgroup]]s B. These turn out to be maximal as connected solvable subgroups (i.e., subgroups with [[composition series]] having as factors one-dimensional subgroups, all of which are groups of additive or multiplicative type); and also minimal such that G/B is a [[projective variety]].
 
The most important subgroups of a linear algebraic group, besides its Borel subgroups, are its [[Algebraic torus|tori]], especially the maximal ones (similar to the study of [[Maximal torus|maximal tori in Lie groups]]). If there is a maximal torus which ''splits'' (i.e. is isomorphic to a product of multiplicative groups), one calls the linear group ''split'' as well. If there is no splitting maximal torus, one studies the splitting tori and the maximal ones of them. If there is a rank at least 1 split torus in the group, the group is called ''isotropic'' and ''anisotropic'' if this is not the case. Any anisotropic or isotropic linear algebraic group over a field becomes split over the [[algebraic closure]], so this distinction is interesting from the point of view of [[Algebraic number theory]].
 
== Group actions ==
Let ''G'' be a [[unipotent group]] acting on an affine variety. Then every ''G''-orbit in the variety is closed.
 
The [[Borel fixed-point theorem]] states that a connected solvable group acting on a non-empty [[complete variety]] admits a fixed point. The classical [[Lie–Kolchin theorem]] follows from the theorem applied to the flag variety.
 
==Non-algebraic Lie groups==
There are several classes of examples of Lie groups that aren't the real or complex points of an algebraic group.
 
*Any Lie group with an infinite group of components G/G<sup>o</sup> cannot be realized as an algebraic group (see [[identity component]]).
*The center of a linear algebraic group is again a linear algebraic group.  Thus, any group whose center has infinitely many components is not a linear algebraic group.  An interesting example is the [[universal cover]] of [[SL2(R)|SL<sub>2</sub>('''R''')]]. This is a Lie group that maps infinite-to-one to SL<sub>2</sub>('''R'''), since the [[fundamental group]] is here [[infinite cyclic]] - and in fact the cover has no faithful matrix representation.
*The general [[solvable Lie group]] need not have a group law expressible by polynomials.
 
==See also==
*[[Differential Galois theory]]
*[[Group of Lie type]] is a group of rational points of a linear algebraic group.
*[[Root system]]s are used in the classification theory of ([[Reductive group|reductive]]) linear algebraic groups.
 
==References==
*[[Armand Borel|Borel, Armand]]. Linear Algebraic Groups (2nd ed.). New York: Springer-Verlag. ISBN 0-387-97370-2.
*[[Brian Conrad|Conrad, Brian]]. [http://math.stanford.edu/~conrad/papers/luminysga3.pdf Reductive group schemes] - develops the theory from the scheme-theoretic viewpoint.
*{{Citation | last1=Kolchin | first1=E. R. | title=Algebraic matric groups and the Picard-Vessiot theory of homogeneous linear ordinary differential equations | jstor=1969111 | mr=0024884 | year=1948 | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=49 | pages=1–42}}
*T. A. Springer, Linear algebraic groups (2nd ed.), Birkhauser, New York, 1998.
 
==External links==
* {{springer|title=Linear algebraic group|id=p/l059070}}
 
[[Category:Algebraic groups]]

Latest revision as of 22:24, 31 October 2014

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