On Multicommutators for Simple Algebraic Groups

On Multicommutators for Simple Algebraic Groups

There are several examples of groups for which any pair of commutators can be written such that both of them have a common entry, and one can look for a similar property for n-tuples of commutators. We here answer, for simple algebraic groups over any field, the weaker question, under which condition the set of n-tuples of commutators with one common entry is Zariski dense in the set of all n-t...

An Invariant of Simple Algebraic Groups

The Rost invariant associated with a simple simply connected algebraic group G is used to define an invariant of strongly inner forms of G. This invariant takes values in a quotient of H(k, Q/Z(2)). It is used to answer a question of Serre about groups of type E6 and to prove a generalization of Gille’s splitting criterion for groups of type E6 and E7. For G a semisimple (affine) algebraic grou...

On the Exponential Map of Almost Simple Real Algebraic Groups

One of the basic problems for Lie groups is to describe the image of the exponential map. A Lie group H is called exponential resp. weakly exponential if its exponential map is surjective resp. has dense image in H . In this paper we classify all exponential groups which are identity components of almost simple real algebraic groups. The weakly exponential groups of this type were determined in...

On Extensions of Simple Modules over Symmetric and Algebraic Groups

The Cohomology of Small Irreducible Modules for Simple Algebraic Groups

LetG be a quasisimple, connected, and simply connected algebraic group defined and split over the field k of characteristic p > 0. In this paper, we are interested in small modules for G; for us, small modules are those with dimension ≤ p. By results of Jantzen [Jan96] one knows that anyGmodule V with dimV ≤ p is semisimple. (We always understand aG-module V to be given by a morphism of algebra...