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The shape of a flat‐faceted octahedral crystal can be uniquely defined by the measured distances between pairs of its parallel facets and the length of one of its false edges. In total, only five numerical values are involved in this approach. Some interdependencies of parameters that allow one to control the correctness of measurements were derived. The proposed method is suitable for describing the shape as full‐faceted, or as incomplete octahedral crystals (e.g. diamond) with unequally developed facets. This so‐called 'real crystal form' can be considered as one of the typomorphic features of minerals, connecting the dissymmetry to the anisotropy of the host rock. The measurement results can be used in crystallo‐morphological analysis, restoration of the lost crystal shape in the case of man‐made damage and in the practice of diamond prospecting. [ABSTRACT FROM AUTHOR]

Let g be a real form of a contragredient Lie superalgebra. Let Aut (g) be the group of g -automorphisms, and let Int (g) be its identity component. We define the outer automorphism group of g by Out (g) = Aut (g) / Int (g). The Kac diagrams are generalizations of Dynkin diagrams, and they represent the even part g 0 ¯ . The Cartan automorphisms are generalizations of Cartan involutions, and they represent g. We express Out (g) in terms of diagram symmetries on the Kac diagrams and Cartan automorphisms. As a result, we obtain Out (g) for all real forms of contragredient Lie superalgebras. [ABSTRACT FROM AUTHOR]

We obtain a necessary and sufficient condition for the existence of equivariant real structures on complex symmetric spaces for semisimple algebraic groups and discuss how to determine the number of equivalence classes for such structures. [ABSTRACT FROM AUTHOR]

This paper is a review of results on the structure of homogeneous ind-varieties G/P of the ind-groups G = GL∞(ℂ), SL∞(ℂ), SO∞(ℂ), and Sp∞(ℂ), subject to the condition that G/P is the inductive limit of compact homogeneous spaces Gn/Pn. In this case, the subgroup P ⊂ G is a splitting parabolic subgroup of G and the ind-variety G/P admits a "flag realization." Instead of ordinary flags, one considers generalized flags that are, in general, infinite chains C of subspaces in the natural representation V of G satisfying a certain condition; roughly speaking, for each nonzero vector 𝜐 of V , there exist the largest space in C , which does not contain 𝜐, and the smallest space in C , which contains v. We start with a review of the construction of ind-varieties of generalized flags and then show that these ind-varieties are homogeneous ind-spaces of the form G/P for splitting parabolic ind-subgroups P ⊂ G. Also, we briefly review the characterization of more general, i.e., nonsplitting, parabolic ind-subgroups in terms of generalized flags. In the special case of the ind-grassmannian X, we give a purely algebraic-geometric construction of X. Further topics discussed are the Bott–Borel–Weil theorem for ind-varieties of generalized flags, finite-rank vector bundles on ind-varieties of generalized flags, the theory of Schubert decomposition of G/P for arbitrary splitting parabolic ind-subgroups P ⊂ G, as well as the orbits of real forms on G/P for G = SL∞(ℂ). [ABSTRACT FROM AUTHOR]

We recall Cartan's definition of characters of real forms of complex simple Lie algebras, based on Cartan decomposition. For a given complex simple Lie algebra, its real forms are uniquely determined by their characters in almost all cases. We work out the exceptions where non-isomorphic real forms have the same character. [ABSTRACT FROM AUTHOR]

We show a necessary and sufficient condition that the fixed point set of a holomorphic isometry and the intersection of two real forms of a Hermitian symmetric space of compact type are discrete and prove that they are antipodal sets in the cases. We also consider some relations between the intersection of two real forms and the fixed point set of a certain holomorphic isometry. [ABSTRACT FROM AUTHOR]

Abstract: We give a new, symmetric construction of the compact real form of the complex Lie algebra . In this construction, the Lie product is determined using an irreducible subgroup of shape ▪ of the automorphism group of . [Copyright &y& Elsevier]

Abstract: Let be the algebra of bounded operators on a complex Banach space X. Viewing as an algebra over , we study the structure of those irreducible subalgebras which contain nonzero compact operators. In particular, irreducible algebras of trace-class operators with real trace are characterized. This yields an extension of Brauer-type results on matrices to operators in infinite dimensions, answering the question: is an irreducible semigroup of compact operators with real spectra realizable, i.e., simultaneously similar to a semigroup whose matrices are real? [Copyright &y& Elsevier]

We show that the superanalogue of the Iwasawa decomposition exists for supergroup SL(n,m,ℂ). The first component of the decomposition is the compact real form SU(n,m), which was constructed following the idea of our article [3]. The second component is the super version of the AN group, that we define in this article. [ABSTRACT FROM AUTHOR]

Abstract: We determine all the Klein surfaces which have a cyclic automorphism group of the maximum possible order, and find their topological types. We also compute their full automorphism groups and show that, except for a finite number of exceptions, they coincide with the full automorphism groups of their Riemann double covers. Explicit algebraic equations of the surfaces and the formulae of their real forms and automorphisms are also given. [Copyright &y& Elsevier]