103,444 research outputs found
Geometry over composition algebras : projective geometry
The purpose of this article is to introduce projective geometry over
composition algebras : the equivalent of projective spaces and Grassmannians
over them are defined. It will follow from this definition that the projective
spaces are in correspondance with Jordan algebras and that the points of a
projective space correspond to rank one matrices in the Jordan algebra. A
second part thus studies properties of rank one matrices. Finally, subvarieties
of projective spaces are discussed.Comment: 24 page
Projective geometry for blueprints
In this note, we generalize the Proj-construction from usual schemes to blue
schemes. This yields the definition of projective space and projective
varieties over a blueprint. In particular, it is possible to descend closed
subvarieties of a projective space to a canonical F_1-model. We discuss this
explicitly in case of the Grassmannian Gr(2,4).Comment: 5 pages, 1 figur
Metric projective geometry, BGG detour complexes and partially massless gauge theories
A projective geometry is an equivalence class of torsion free connections
sharing the same unparametrised geodesics; this is a basic structure for
understanding physical systems. Metric projective geometry is concerned with
the interaction of projective and pseudo-Riemannian geometry. We show that the
BGG machinery of projective geometry combines with structures known as
Yang-Mills detour complexes to produce a general tool for generating invariant
pseudo-Riemannian gauge theories. This produces (detour) complexes of
differential operators corresponding to gauge invariances and dynamics. We
show, as an application, that curved versions of these sequences give geometric
characterizations of the obstructions to propagation of higher spins in
Einstein spaces. Further, we show that projective BGG detour complexes generate
both gauge invariances and gauge invariant constraint systems for partially
massless models: the input for this machinery is a projectively invariant gauge
operator corresponding to the first operator of a certain BGG sequence. We also
connect this technology to the log-radial reduction method and extend the
latter to Einstein backgrounds.Comment: 30 pages, LaTe
Projective Cross-Ratio on Hypercomplex Numbers
The paper presents a new cross-ratio of hypercomplex numbers based on
projective geometry. We discuss the essential properties of the projective
cross-ratio, notably its invariance under Mobius transformations. Applications
to the geometry of conic sections and Mobius-invariant metrics on the upper
half-plane are also given
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