Embeddings and hyperplanes of the Lie incidence geometry of type \(E_{7,1}\)
type: Item
instance of: scholarly article
title: Embeddings and hyperplanes of the Lie incidence geometry of type \(E_{7,1}\)
zbMATH Open document ID: 0885.51005
DOI: 10.1007/BF01229573
published in: Journal of Geometry
publication date: 1998-04-07T00:00:00Z
review text: A Lie incidence geometry is a point-line geometry \(\Gamma=({\mathcal P},{\mathcal L})\) whose points are the cosets of a maximal parabolic subgroup \(P\) of a group of Lie type and whose lines are the cosets of the parabolic subgroup corresponding to the collection of all nodes in the Dynkin diagram adjacent to the unique node corresponding to \(P\). A (projective) embedding \(e\) of \(\Gamma\) is a one-to-one map from \({\mathcal P}\) into the point set of some projective space \({\mathbf P}(V)\) over a vector space \(V\) such that \(e(L)\) is a projective line for each line \(L\in{\mathcal L}\) and such that the image points \(e({\mathcal P})\) span \({\mathbf P}(V)\). A geometric hyperplane \(H\) of \(\Gamma\), that is, a proper subspace which meets each line non-trivially, is said to arise from \(e\) if \(H=e^{-1}(H')\) for a projective hyperplane \(H'\) of \({\mathbf P}(V)\). If every geometric hyperplane of a Lie incidence geometry \(\Gamma\) arises from an embedding \(e\), then \(e\) is relatively universal, that is, \(e\) is a source in the subcategory of all projective embeddings of \(\Gamma\) possessing a morphism to \(e\). The question whether or not every geometric hyperplane of a Lie incidence geometry arises from an embedding has been answered in the affirmative for a large class of Lie incidence geometries by the author and others in a series of papers. For example, projective spaces, polar spaces of rank at least 2, Grassmann spaces of type \(A_{n,k}\) over a field, spin geometries over finite fields of odd order, half-spin geometries, geometries of type \(D_{n,n}\) and the exceptional geometries of type \(E_{6,1}\) are known to fall into this class. In the paper under review the author adds Lie incidence geometries of type \(E_{7,1}\) to this list. More precisely, he shows that for any relative universal embeddding \(e\) every geometric hyperplane arises from \(e\). Furthermore, such an embedding is unique and hence absolutely universal in the cate…
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 51B25
zbMATH DE Number: 1053897
zbMATH Keywords: embedding
zbMATH Keywords: geometric hyperplane
zbMATH Keywords: Lie incidence geometry
zbMATH Keywords: parapolar space
author: E. E. Shult
reviewed by: Günter F. Steinke
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1053897