Linear Topological Geometries

Grundhöfer, Theo; Löwen, Rainer

doi:10.1016/b978-044488355-1/50025-6

Cited by 16 publications

(4 citation statements)

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“…This face contains F properly (because it contains p) so h ′ < h + 1. Now recall that a topological projective space is a projective space in which the sets of flats of each rank are given nontrivial topologies that make the join and meet operations ∨ and ∧ continuous, when restricted to pairs of flats of fixed ranks whose join or meet have a fixed rank [5]. Lemma 3.…”

Section: Face Lattices Defining Projective Spacesmentioning

confidence: 99%

Cones and convex bodies with modular face lattices

Labardini-Fragoso

Neumann-Coto

Takane

2012

Proc. Amer. Math. Soc.

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Dedicated to Claus M. Ringel on the occasion of his 60th birthday.Abstract. If a convex body C has modular and irreducible face lattice (and is not strictly convex), there is a face-preserving homeomorphism from C to a section of a cone of hermitian matrices over R, C, or H, or C has dimension 8, 14 or 26.1991 Mathematics Subject Classification. 52A20, 06C05, 51A05, 15A48.Recall that a projective space consists of a set P (the points) and a set L (the lines) so that (a) Each pair of points is contained in a unique line, (b) If a, b, c, d are distinct points and the lines ab and cd intersect, then the lines ac and bd intersect (c) Each line contains at least 3 points and there are at least 2 lines (d) Every chain of subspaces (also called flats) has finite length. The maximum length of a chain starting with a point is the projective dimension of the space.The flats of a projective space form an algebraic, atomic, irreducible, modular lattice. Conversely, any lattice with these properties is the lattice of flats of a

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Section: Face Lattices Defining Projective Spacesmentioning

confidence: 99%

Cones and convex bodies with modular face lattices

Labardini-Fragoso

Neumann-Coto

Takane

2012

Proc. Amer. Math. Soc.

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“…Q. 3 Often we use the notations from the classical line geometry: a union of an n-parameter family of lines, n ∈ {1, 2, 3}, is called for n = 1 a ruled surface, for n = 2 a congruence of lines, and for n = 3 a complex of lines. …”

Section: Construct Examples Of Topological Non-axial Generalized Linementioning

confidence: 99%

Generalized Line Stars and Topological Parallelisms of the Real Projective 3-Space

Betten

Riesinger²

2008

J. geom.

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Let Q be an elliptic quadric of the real projective 3-space PG(3, R) =: Π3 and denote by Q ¬ i the set of non-interior points with respect to Q. A simple covering S of Q ¬ i by 2-secants of Q is called generalized line star with respect to Q. The generalized line star S is called continuous, if the determination of the unique line of S through a given point of Q ¬ i is continuous. A parallelism is a family P of spreads such that each line of Π3 is contained in exactly one spread of P; two lines of Π3 are P-parallel if, and only if, they are members of the same spread of P. The parallelism P is called topological, if the operation of drawing a line P-parallel to a given line through a given point is continuous. In [1] the authors give a construction P such that P(S) is a parallelism of Π3; cf. Theorem 1.4 below.In the present paper the authors prove: If S is continuous, then P(S) is a topological parallelism of Π3. The authors construct continuous generalized line stars by composing so-called 3-pencils (cf. Definition 3.3 below). If A is a generalized line star of [1, Section 5] or of [2] or from Section 3 of the present paper, then there exists a line A such that each line of A has non-empty intersection with A; we speak of an axial generalized line star. In Section 4 examples of non-axial continuous generalized line stars are constructed.Mathematics Subject Classification (2000). 51H10; 51A40, 51M30.

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“…This follows either from the topological Fundamental Theorem of Projective Geometry, (as proved by Kolmogorov [27], see also Kühne-Löwen [30] and the survey by Grundhöfer-Löwen [17]), or from Cartan's classification of simple Lie groups and Tits' classification [42] of their Tits diagrams (unjustly sometimes called "Satake diagrams") and their relative diagrams, see Helgason [20] Table VI, Ch. X p. 532.…”

Section: Theoremmentioning

confidence: 99%

Two-transitive Lie groups

Krämer¹

2003

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Using a characterization of parabolics in reductive Lie groups due to Furstenberg, elementary properties of buildings, and some algebraic topology, we give a new proof of Tits' classification of 2-transitive Lie groups.Among many other results, Tits classified in [41] all 2-transitive Lie groups. His proof is based on Dynkin's classification of maximal complex subalgebras in complex simple Lie algebras; it is long and depends on consideration of various cases. Since the resulting list of groups is also long (at least in the affine case), it is clear that there cannot be a very short proof of the full classification. On the other hand, Lie theory has developed since the time [41] was written. In particular, Tits himself changed the picture through his theory of buildings (as he pointed out, his paper [41] was one of the motivations for him to invent buildings). The language, the methods, and the terminology have changed since then, and it is natural to look for a new (and shorter) proof of Tits' classification. Note also that the proof presented in [41] IV F 1.2, p. 222, does not cover certain real forms of exceptional groups -a footnote on p. 223 asserts that Tits found a proof for these cases, too, after the manuscript went into print; see also loc.cit. p. 240. The details were never published.Almost at the same time as Tits, Borel [4] determined all 1-connected spaces X which admit a 2-or 3-transitive Lie group action; however, Borel did not classify the corresponding groups. His proof relies on spectral sequences, Freudenthal's theory of ends, and on the results of Borel and De Siebenthal about homogeneous spaces of positive Euler characteristic.In this paper, we give a complete proof for Tits' classification. The main ingredients are a characterization of parabolics in Lie groups due to Furstenberg, elementary properties of buildings, some algebraic topology (certainly more elementary than the machinery employed in Borel's work [4]), and representation theory of semisimple (compact) Lie groups. * Supported by a Heisenberg fellowship by the Deutsche Forschungsgemeinschaft 1 As we remarked before, the only published proof for the classification is [41]. The classification in the affine case is also stated (but not proved) in Völklein [47], and some remarks on the strategy of Tits' original proof can be found in Salzmann et al. [36] 96.15 and 96.16.Related results for other classes of groups are Knop's classification [26] of 2-transitive actions of algebraic groups over algebraically closed fields in arbitrary characteristic (which is achieved by quite different methods), and the classification of all finite 2-transitive groups; see Dixon-Mortimer [11] 7.7 for a description of these groups. In the course of our classification, we recover Knop's result for the special case of complex algebraic groups.The main results of the classification are as follows.Theorem A Let G be a locally compact, σ-compact topological transformation group acting effectively and 2-transitively on a space X which is not totally...

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Linear Topological Geometries

Cited by 16 publications

References 261 publications

Cones and convex bodies with modular face lattices

Cones and convex bodies with modular face lattices

Generalized Line Stars and Topological Parallelisms of the Real Projective 3-Space

Two-transitive Lie groups

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