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Difference between revisions of "Anisotropic group"

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''over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a0125301.png" />''
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{{MSC|20.27|}}
  
A [[Linear algebraic group|linear algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a0125302.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a0125303.png" /> and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a0125304.png" />-rank zero, i.e. not containing non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a0125305.png" />-split tori (cf. [[Splittable group|Splittable group]]). Classical examples of anisotropic groups include the orthogonal groups of quadratic forms that do not vanish over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a0125306.png" />; and algebraic groups of elements of reduced norm one in division algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a0125307.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a0125308.png" /> is semi-simple, and if the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a0125309.png" /> is zero, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a01253010.png" /> is anisotropic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a01253011.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a01253012.png" /> contains non-trivial unipotent elements. (For the field of real numbers or the field of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a01253013.png" />-adic numbers this is equivalent to saying that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a01253014.png" /> is compact.) The classification of arbitrary semi-simple groups over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012530/a01253015.png" /> reduces essentially to the classification of anisotropic groups.
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An anisotropic algebraic group over a field $k$ is a
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[[Linear algebraic group|linear algebraic group]] $G$ defined over $k$  [1]
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and of $k$-rank zero, i.e. not containing non-trivial $k$-split tori
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(cf.
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[[Splittable group|Splittable group]]). Classical examples of
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anisotropic groups include the orthogonal groups of quadratic forms
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that do not vanish over $k$; and algebraic groups of elements of
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reduced norm one in division algebras over $k$. If $G$ is semi-simple,
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and if the characteristic of $k$ is zero, then $G$ is anisotropic over
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$k$ 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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel,   "Linear algebraic groups" , Benjamin (1969)</TD></TR><TR><TD valign="top">[2]</TD> <TD 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</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD>
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<TD valign="top"> A. Borel, ''Linear algebraic groups'', Benjamin (1969) | {{MR|0251042}} | {{ZBL|0186.33201}} </TD>
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</TR><TR><TD valign="top">[2]</TD>
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<TD valign="top"> 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 | {{MR|0224710}} | {{ZBL|0238.20052 }} </TD>
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</TR></table>

Revision as of 22:56, 23 December 2011

2010 Mathematics Subject Classification: Primary: 20.27 [MSN][ZBL]

An anisotropic algebraic group over a field $k$ is a linear algebraic group $G$ defined over $k$ [1] and of $k$-rank zero, i.e. not containing non-trivial $k$-split tori (cf. 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 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=14493
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article