264 research outputs found
Automorphisms and opposition in twin buildings
We show that every automorphism of a thick twin building interchanging the
halves of the building maps some residue to an opposite one. Furthermore we
show that no automorphism of a locally finite 2-spherical twin building of rank
at least 3 maps every residue of one fixed type to an opposite. The main
ingredient of the proof is a lemma that states that every duality of a thick
finite projective plane admits an absolute point, i.e., a point mapped onto an
incident line. Our results also hold for all finite irreducible spherical
buildings of rank at least 3, and as a consequence we deduce that every
involution of a thick irreducible finite spherical building of rank at least 3
has a fixed residue
Automorphisms of classical geometries in the sense of Klein
In this note, we compute the group of automorphisms of Projective, Affine and
Euclidean Geometries in the sense of Klein.
As an application, we give a simple construction of the outer automorphism of
S_6.Comment: 8 page
On Deriving Space-Time From Quantum Observables and States
We prove that, under suitable assumptions, operationally motivated data
completely determine a space-time in which the quantum systems can be
interpreted as evolving. At the same time, the dynamics of the quantum system
is also determined. To minimize technical complications, this is done in the
example of three-dimensional Minkowski space.Comment: 19 pages, to appear in Communications in Mathematical Physics; minor
corrections mad
Collineations of smooth stable planes
Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)Smooth stable planes have been introduced in [4]. We show that every continuous collineation between two smooth stable planes is in fact a smooth collineation. This implies that the group Î of all continuous collineations of a smooth stable plane is a Lie transformation group on both the set P of points and the set â of lines. In particular, this shows that the point and line sets of a (topological) stable plane â admit at most one smooth structure such that â becomes a smooth stable plane. The investigation of central and axial collineations in the case of (topological) stable planes due to R. Löwen ([25], [26], [27]) is continued for smooth stable planes. Many results of [26] which are only proved for low dimensional planes (dim â †4) are transferred to smooth stable planes of arbitrary finite dimension. As an application of these transfers we show that the stabilizers Î[c,c] 1 and Î[A,A] 1 (see (3.2) Notation) are closed, simply connected, solvable subgroups of Aut(â) (Corollary (4.17)). Moreover, we show that Î[c,c] is even abelian (Theorem (4.18)). In the closing section we investigate the behaviour of reflections
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