Anisotropic kernel

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