Kimura, T.; Kasai, S.; Hosokawa, H. Universal transitivity of simple and 2-simple prehomogeneous vector spaces. (English) Zbl 0606.14037 Ann. Inst. Fourier 38, No. 2, 11-41 (1988). We denote by k a field of characteristic zero satisfying \(H^ 1(k,Aut(SL_ 2))\neq 0\). Let G be a connected k-split linear algebraic group acting on \(X=Aff^ n\) rationally by \(\rho\) with a Zariski-dense G- orbit Y. A prehomogeneous vector space (G,\(\rho\),X) is called ”universally transitive” if the set of k-rational points Y(k) is a single \(\rho\) (G)(k)-orbit for all such k. Such prehomogeneous vector spaces are classified by J. Igusa when \(\rho\) is irreducible. We classify them when G is reductive and its commutator subgroup [G,G] is either a simple algebraic group or a product of two simple algebraic groups. Cited in 1 ReviewCited in 3 Documents MSC: 14M17 Homogeneous spaces and generalizations 12G05 Galois cohomology 20G15 Linear algebraic groups over arbitrary fields Keywords:prehomogeneous vector spaces; Galois cohomology PDFBibTeX XMLCite \textit{T. Kimura} et al., Ann. Inst. Fourier 38, No. 2, 11--41 (1988; Zbl 0606.14037) Full Text: DOI Numdam EuDML References: [1] [1] , On functional equations of complex powers, Invent. Math., 85 (1986), 1-29. · Zbl 0599.12017 [2] [2] , On a certain class of prehomogeneous vector spaces, to appear in Journal of Algebra. · Zbl 0765.14013 [3] [3] and , A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J., 65 (1977), 1-155. · Zbl 0321.14030 [4] [E] , Periodic flows on 3-manifolds, Ann. of Math., 95 (1972), 68-92. · Zbl 0533.14024 [5] [5] , , and , A classification of 2-simple prehomogeneous vector spaces of type I, to appear in Journal of Algebra. · Zbl 0658.20026 [6] [6] , , and , Some P.V.-equivalences and a classification of 2-simple prehomogeneous vector spaces of type II, to appear in Transaction of A.M.S. · Zbl 0666.14021 [7] [7] , Cohomologie Galoisienne, Springer Lecture Note, 5 (1965). · Zbl 0136.02801 [8] [8] , Formes réelles des espaces préhomogènes irréductibles de type parabolique, Annales de l’Institut Fourier, Grenoble, 36-1 (1986), 1-38. · Zbl 0588.17007 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.