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Any complex-analytic vector bundle E admits naturally defined homotheties ϕ α , α ∈ C ⁎ , i.e. ϕ α is the multiplication of a local section by a complex number α. We investigate the question when such automorphisms can be lifted to a non-split supermanifold corresponding to E. Further, we compute the automorphism supergroup of a Π-symmetric super-Grassmannian Π Gr n , k , and, using Galois cohomology, we classify the real structures on Π Gr n , k and compute the corresponding supermanifolds of real points. [ABSTRACT FROM AUTHOR]

In this paper we classify real trivectors in dimension 9. The corresponding classification over the field C of complex numbers was obtained by Vinberg and Elashvili in 1978. One of the main tools used for their classification was the construction of the representation of SL (9 , C) on the space of complex trivectors of C 9 as a theta-representation corresponding to a Z / 3 Z -grading of the simple complex Lie algebra of type E 8. This divides the trivectors into three groups: nilpotent, semisimple, and mixed trivectors. Our classification follows the same pattern. We use Galois cohomology, first and second, to obtain the classification over R. [ABSTRACT FROM AUTHOR]

Let G be a simply connected absolutely simple algebraic group defined over the field of real numbers R . Let H be a simply connected semisimple R -subgroup of G . We consider the homogeneous space X = G / H . We ask: how many connected components has X ( R ) ? We give a method of answering this question. Our method is based on our solutions of generalized Reeder puzzles. [ABSTRACT FROM AUTHOR]

A linear algebraic group G over a field K is called a Cayley K -group if it admits a Cayley map, i.e., a G -equivariant K -birational isomorphism between the group variety G and its Lie algebra. We classify real reductive algebraic groups of absolute rank 1 and 2 that are Cayley R -groups. [ABSTRACT FROM AUTHOR]

Let G be a connected linear algebraic group over a geometric field k of cohomological dimension 2 of one of the types which were considered by Colliot-The´le`ne, Gille and Parimala. Basing on their results, we compute the group of classes of R-equivalence G(k)/R, the defect of weak approximation AΣ(G), the first Galois cohomology H1(k,G), and the Tate–Shafarevich kernel ш1(k,G) (for suitable k) in terms of the algebraic fundamental group π1(G). We prove that the groups G(k)/R and AΣ(G) and the set ш1(k,G) are stably k-birational invariants of G. [Copyright &y& Elsevier]