20 research outputs found
Abstract involutions of algebraic groups and of Kac-Moody groups
Based on the second author's thesis in this article we provide a uniform
treatment of abstract involutions of algebraic groups and of Kac-Moody groups
using twin buildings, RGD systems, and twisted involutions of Coxeter groups.
Notably we simultaneously generalize the double coset decompositions
established by Springer and by Helminck-Wang for algebraic groups and by
Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by
Devillers-Muhlherr in the context of arbitrary involutions, and we answer a
structural question on the combinatorics of involutions of twin buildings
raised by Bennett-Gramlich-Hoffman-Shpectorov
Iwasawa decompositions of split Kac-Moody groups
We characterize all fields F for which a group with an F-locally split root
group datum admits an Iwasawa decomposition. This class of groups in particular
includes the split semisimple algebraic groups and the split Kac-Moody groups
Moufang sets of type F-4
We give an explicit description of the Moufang sets of type F-4, i.e. the buildings arising from the simple algebraic groups of absolute type F4 and relative rank one, over an arbitrary field. We use octonion planes and certain polarities to find this description, and we rely on the theory of Albert algebras. We also determine the automorphism groups of the corresponding exceptional unitals, thereby completing the program of J. Tits for these non-abelian Moufang sets. In particular we prove that every automorphism of that unital is induced by a collineation of the ambient projective plane