Homomorphisms from linear groups over division rings to algebraic groups

1986 ◽  
pp. 231-265 ◽  
Author(s):  
Yu Chen
Keyword(s):  
Algebraic Groups ◽  
Linear Groups ◽  
Division Rings

scholarly journals Équivalence motivique des groupes algébriques semisimples

2017 ◽  
Vol 153 (10) ◽  
pp. 2195-2213
Author(s):  
Charles De Clercq
Keyword(s):  
Algebraic Group ◽  
Algebraic Groups ◽  
Linear Groups ◽  
Flag Varieties ◽  
Orthogonal Groups ◽  
Special Linear ◽  
Special Linear Groups ◽  
Field Extensions ◽  
Semisimple Algebraic Groups ◽  
Partial Version

We prove that the standard motives of a semisimple algebraic group$G$with coefficients in a field of order$p$are determined by the upper motives of the group $G$. As a consequence of this result, we obtain a partial version of the motivic rigidity conjecture of special linear groups. The result is then used to construct the higher indexes which characterize the motivic equivalence of semisimple algebraic groups. The criteria of motivic equivalence derived from the expressions of these indexes produce a dictionary between motives, algebraic structures and the birational geometry of twisted flag varieties. This correspondence is then described for special linear groups and orthogonal groups (the criteria associated with other groups being obtained in De Clercq and Garibaldi [Tits$p$-indexes of semisimple algebraic groups, J. Lond. Math. Soc. (2)95(2017) 567–585]). The proofs rely on the Levi-type motivic decompositions of isotropic twisted flag varieties due to Chernousov, Gille and Merkurjev, and on the notion of pondered field extensions.

Unipotent normal subgroups of skew linear groups

1989 ◽  
Vol 105 (1) ◽  
pp. 37-51 ◽  
Author(s):  
B. A. F. Wehrfritz
Keyword(s):  
Normal Subgroup ◽  
Matrix Group ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Matrix Groups ◽  
Division Rings ◽  
Additional Assumptions

A recurrent problem over many years in the study of linear groups has been the determination of the central height of a unipotent normal subgroup of some matrix group of specified type. In the theory of matrix groups over division rings, unipotent elements frequently present special difficulties and these have usually been by-passed by the addition of some suitable hypothesis. In this paper we make a start on the removal of these extraneous hypotheses. Our motivation for doing this now conies from [9], where by 3·7 of that paper the additional assumptions have finally reduced us to degree one, a situation where unipotent elements present few problems!

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