The classical geometries of points and lines include not only the projective and polar spaces, but similar truncations of geometries naturally arising from the groups of Lie type. Virtually all of these geometries (or homomorphic images of them) are characterized in this book by simple local axioms on points and lines. Simple point-line characterizations of Lie incidence geometries allow one to recognize Lie incidence geometries and their automorphism groups. These tools could be useful in shortening the enormously lengthy classification of finite simple groups. Similarly, recognizing ruled manifolds by axioms on light trajectories offers a way for a physicist to recognize the action of a Lie group in a context where it is not clear what Hamiltonians or Casimir operators are involved. The presentation is self-contained in the sense that proofs proceed step-by-step from elementary first principals without further appeal to outside results. Several chapters have new heretofore unpublished research results. On the other hand, certain groups of chapters would make good graduate courses. All but one chapter provide exercises for either use in such a course, or to elicit new research directions.
Author(s): Ernest Shult (auth.)
Series: Universitext
Edition: 1
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2011
Language: English
Pages: 676
Tags: Geometry; Topological Groups, Lie Groups
Front Matter....Pages i-xxii
Front Matter....Pages 1-1
Basics About Graphs....Pages 3-41
Geometries: Basic Concepts....Pages 43-58
Point-Line Geometries....Pages 59-77
Hyperplanes, Embeddings, and Teirlinck’s Theory....Pages 79-101
Front Matter....Pages 103-103
Projective Planes....Pages 105-133
Projective Spaces....Pages 135-165
Polar Spaces....Pages 167-249
Near Polygons....Pages 251-287
Front Matter....Pages 289-289
Chamber Systems and Buildings....Pages 291-397
2-Covers of Chamber Systems....Pages 399-413
Locally Truncated Diagram Geometries....Pages 415-440
Separated Systems of Singular Spaces....Pages 441-453
Cooperstein’s Theory of Symplecta and Parapolar Spaces....Pages 455-494
Front Matter....Pages 495-495
Characterizations of the Classical Grassmann Spaces....Pages 497-526
Characterizing the Classical Strong Parapolar Spaces: The Cohen–Cooperstein Theory Revisited....Pages 527-552
Characterizing Strong Parapolar Spaces by the Relation Between Points and Certain Maximal Singular Subspaces....Pages 553-601
Point-Line Characterizations of the “Long Root Geometries”....Pages 603-626
The Peculiar Pentagon Property....Pages 627-663
Back Matter....Pages 665-676
