Difference between revisions of "Anisotropic group"
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| − | An anisotropic algebraic group over a field $k$ is a | + | An anisotropic algebraic group over a [[Field|field]] $k$ is a |
| − | [[Linear algebraic group|linear algebraic group]] $G$ defined over $k$ | + | [[Linear algebraic group|linear algebraic group]] $G$ defined over $k$ |
| − | and of $k$-rank zero, i.e. not containing non-trivial $k$-split tori | + | and of $k$-rank zero, i.e. not containing non-trivial [[Algebraic torus|$k$-split tori]] [1]. Classical examples of |
| − | + | anisotropic groups include the [[Orthogonal group|orthogonal groups]] of [[Quadratic form|quadratic forms]] | |
| − | [ | ||
| − | 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, | + | [[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 | + | $k$ 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 | + | 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 |
Anisotropic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anisotropic_group&oldid=19931