Difference between revisions of "Anisotropic kernel"
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| − | The subgroup | + | {{TEX|done}} |
| + | The subgroup $ D $ | ||
| + | of a semi-simple [[Algebraic group|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|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 $ [[#References|[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==== | ====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> | <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=21851
Anisotropic kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anisotropic_kernel&oldid=21851
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article