Chain spaces via Clifford algebras
The author continues her investigation of chain spaces. She considers the chain space \(\Sigma({\mathcal Q})\) associated with a proper non-degenerate quadric \({\mathcal Q}\) in projective space of dimension at least 3 over a commutative field \(K\). The point set of \(\Sigma({\mathcal Q})\) contains all points of \({\mathcal Q}\) except its double points and the chain set consists of all non-trivial plane sections of \({\mathcal Q}\). This generalises the familiar models of the Miquelian Möbius, Laguerre and Minkowski planes in case \(K\) being the field of reals and \({\mathcal Q}\) being an elliptic quadric, an elliptic cylinder or a ruled quadric in 3-dimensional real projective space, respectively. Another algebraic description of these three Miquelian Benz planes is one as a chain geometry belonging to the 2-dimensional real kinematic algebras, that is, the complex numbers, the dual numbers and the anomal complex numbers in the respective cases. The author extends the algebraic descriptions of Miquelian Benz planes to one for chain spaces \(\Sigma({\mathcal Q})\) in case \(K\) has at least 5 elements. The underlying vector space \(W\) can be written as the orthogonal sum of a metric vector space \(V\) with quadratic form \(Q\) and a hyperbolic plane. Let \(A\) be the Clifford algebra associated with \((V,Q)\) and let \(B=Vw\) for some vector \(w\in V\) with \(Q(w)=1\). Then \(B\) is a strong Jordan system in \(A\) and gives rise to a subspace \(\Sigma(K,A,B)\) of the chain geometry \(\Sigma(K,A)\) (the projective line over \(A\) with chains being the images of the projective line over \(K\) under the general linear group over \(A\)). In fact, up to isomorphism, \(\Sigma(K,A,B)\) is independent of the choice of \(w\) since different vectors yield isotopic Jordan systems and these in turn yield isomorphic chain spaces, see the author [Abh. Math. Semin. Univ. Hamb. 64, 33-49 (1994; Zbl 0817.51004)]. It is shown that \(\Sigma({\mathcal Q})\) and \(\Sigma(K,A,B)\) are isomorphic chain spaces. This is achieved by specifying an isomorphism. In fact, this isomorphism can be interpreted as an extended stereographic projection from a point of the quadric \({\mathcal Q}\) and is an explicit exposition of an isomorphism proof via the reconstruction functor for sketched geometries described by \textit{M. Stroppel} [Geom. Dedicata 46, 47-60 (1993; Zbl 0783.51002)]. In particular, almost every chain space \(\Sigma({\mathcal Q})\) on a quadric \({\mathcal Q}\) can be embedded into a chain geometry \(\Sigma(K,A)\). This result partially answers a question posed by \textit{A. Herzer} in [Chain geometries, Handbook of incidence geometry: buildings and foundations. Amsterdam: North-Holland. pp. 781-842 (1995; Zbl 0829.51003)].
scientific article; zbMATH DE number 986868
scientific article