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The subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125401.png" /> of a semi-simple [[Algebraic group|algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125402.png" />, defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125403.png" />, which is the commutator subgroup of the centralizer of a maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125404.png" />-split torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125405.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125406.png" />. The anisotropic kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125407.png" /> is a semi-simple [[Anisotropic group|anisotropic group]] defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125408.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a0125409.png" />. The concept of the anisotropic kernel plays an important role in the study of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254010.png" />-structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254011.png" /> [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254012.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254014.png" /> is anisotropic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254015.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254016.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254017.png" /> is called quasi-split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012540/a01254018.png" />.
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The subgroup $  D $
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of a semi-simple [[Algebraic group|algebraic group]] $  G $ ,  
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defined over a field $  k $ ,  
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which is the commutator subgroup of the centralizer of a maximal $  k $ -
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split torus $  S \subset G $ ;  
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$  D = [ {Z _{G}} (S),\  {Z _{G}} (S)] $ .  
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The anisotropic kernel $  D $
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is a semi-simple [[Anisotropic group|anisotropic group]] defined over $  k $ ;  
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$  { \mathop{\rm rank}\nolimits} \  D = { \mathop{\rm rank}\nolimits} \  G - { \mathop{\rm rank}\nolimits _{k}} \  G $ .  
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The concept of the anisotropic kernel plays an important role in the study of the $  k $ -
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structure of $  G $ [[#References|[1]]]. If $  D = G $ ,  
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i.e. if $  { \mathop{\rm rank}\nolimits _{k}} \  G = 0 $ ,  
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then $  G $
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is anisotropic over $  k $ ;  
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if $  D = (e) $ ,  
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the group $  G $
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is called quasi-split over $  k $ .
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====References====
 
====References====
<table><TR><TD valign="top">[1]</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 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Borel,  J. Tits,  "Groupes réductifs"  ''Publ. Math. IHES'' , '''27'''  (1965)  pp. 55–150  {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR></table>
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<table><TR><TD valign="top">[1]</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   {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Borel,  J. Tits,  "Groupes réductifs"  ''Publ. Math. IHES'' , '''27'''  (1965)  pp. 55–150  {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR></table>

Latest revision as of 10:43, 17 December 2019

The subgroup $ D $ of a semi-simple algebraic group $ G $ , defined over a field $ k $ , which is the commutator subgroup of the centralizer of a maximal $ k $ - split torus $ S \subset G $ ; $ D = [ {Z _{G}} (S),\ {Z _{G}} (S)] $ . The anisotropic kernel $ D $ is a semi-simple anisotropic group defined over $ k $ ; $ { \mathop{\rm rank}\nolimits} \ D = { \mathop{\rm rank}\nolimits} \ G - { \mathop{\rm rank}\nolimits _{k}} \ G $ . The concept of the anisotropic kernel plays an important role in the study of the $ k $ - structure of $ G $ [1]. If $ D = G $ , i.e. if $ { \mathop{\rm rank}\nolimits _{k}} \ G = 0 $ , then $ G $ is anisotropic over $ k $ ; if $ D = (e) $ , the group $ G $ is called quasi-split over $ k $ .


References

[1] J. Tits, "Classification of algebraic simple groups" , Algebraic Groups and Discontinuous Subgroups , Proc. Symp. Pure Math. , 9 , Amer. Math. Soc. (1966) pp. 33–62
[2] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
How to Cite This Entry:
Anisotropic kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anisotropic_kernel&oldid=21814
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article