# Anisotropic kernel

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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=44271
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article