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Difference between revisions of "Anisotropic group"

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{{MSC|20.27|}}
 
{{MSC|20.27|}}
  
An anisotropic algebraic group over a field $k$ is a
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An anisotropic algebraic group over a [[Field|field]] $k$ is a
[[Linear algebraic group|linear algebraic group]] $G$ defined over $k$ [1]
+
[[Linear algebraic group|linear algebraic group]] $G$ defined over $k$  
and of $k$-rank zero, i.e. not containing non-trivial $k$-split tori
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and of $k$-rank zero, i.e. not containing non-trivial [[Algebraic torus|$k$-split tori]] [1]. Classical examples of
(cf.
+
anisotropic groups include the [[Orthogonal group|orthogonal groups]] of [[Quadratic form|quadratic forms]]
[[Splittable group|Splittable group]]). Classical examples of
 
anisotropic groups include the orthogonal groups of quadratic forms
 
 
that do not vanish over $k$; and algebraic groups of elements of
 
that do not vanish over $k$; and algebraic groups of elements of
reduced norm one in division algebras over $k$. If $G$ is semi-simple,
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[[Reduced norm|reduced norm]] one in [[Division algebra|division algebras]] over $k$. If $G$ is [[Semi-simple algebraic group|semi-simple]],
 
and if the characteristic of $k$ is zero, then $G$ is anisotropic over
 
and if the characteristic of $k$ is zero, then $G$ is anisotropic over
$k$ if and only if $G_k$ contains non-trivial unipotent elements. (For
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$k$ if and only if $G_k$ contains non-trivial [[Unipotent element|unipotent elements]]. (For
the field of real numbers or the field of $p$-adic numbers this is
+
the field of [[Real number|real numbers]] or the field of [[P-adic number|$p$-adic numbers]] this is
equivalent to saying that $G_k$ is compact.) The classification of
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equivalent to saying that $G_k$ is [[Compact space|compact]].) The classification of
 
arbitrary semi-simple groups over the field $k$ reduces essentially to
 
arbitrary semi-simple groups over the field $k$ reduces essentially to
 
the classification of anisotropic groups over $k$ [2].
 
the classification of anisotropic groups over $k$ [2].

Revision as of 23:55, 23 December 2011

2020 Mathematics Subject Classification: Primary: 20.27 [MSN][ZBL]

An anisotropic algebraic group over a field $k$ is a linear algebraic group $G$ defined over $k$ and of $k$-rank zero, i.e. not containing non-trivial $k$-split tori [1]. Classical examples of anisotropic groups include the orthogonal groups of quadratic forms that do not vanish over $k$; and algebraic groups of elements of reduced norm one in division algebras over $k$. If $G$ is semi-simple, and if the characteristic of $k$ is zero, then $G$ is anisotropic over $k$ if and only if $G_k$ contains non-trivial unipotent elements. (For the field of real numbers or the field of $p$-adic numbers this is equivalent to saying that $G_k$ is compact.) The classification of arbitrary semi-simple groups over the field $k$ reduces essentially to the classification of anisotropic groups over $k$ [2].

References

[1] A. Borel, Linear algebraic groups, Benjamin (1969) | MR0251042 | Zbl 0186.33201
[2] J. Tits, Classification of algebraic semisimple groups, in Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math., 9, Amer. Math. Soc. (1966) pp. 33–62 | MR0224710 | Zbl 0238.20052
How to Cite This Entry:
Anisotropic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anisotropic_group&oldid=19931
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article