Four-dimensional compact projective planes with collineation group \(N_{6,28}\)
type: Item
instance of: scholarly article
title: Four-dimensional compact projective planes with collineation group \(N_{6,28}\)
zbMATH Open document ID: 0838.51004
DOI: 10.1007/BF03322245
author: Hauke Klein
published in: Results in Mathematics
publication date: 1996-06-03T00:00:00Z
review text: This paper deals with the last remaining case in the classification of all flexible 4-dimensional compact projective planes, i.e. topological projective planes with point space homeomorphic to the classical plane over the complex numbers such that the group of continuous collineations has an open orbit in the space of flags. Flexible planes have collineation groups of dimension at least 6. All 4-dimensional compact projective planes admitting a 7-dimensional collineation group have been classified. Up to duality these planes are translation planes or the shift plane of Knarr. Flexible 4-dimensional compact projective planes with 6-dimensional collineation groups have been classified by a number of authors except for one case, which is studied in this paper. It is assumed that the connected component \(\Sigma\) of the group of continuous collineations contains no subgroup isomorphic to the vector group \(\mathbb{R}^4\) and that \(\Sigma\) is solvable. In this case \(\Sigma\) fixes a flag \((v,W)\). It is then further assumed that, up to duality, \(\Sigma\) acts transitively on the line pencil \({\mathcal L}_v \backslash \{W\}\) and fixes no point on \(W\) except \(v\). The author [Geom. Dedicata 58, No. 1, 53-62 (1995); see the paper below] proved under these assumptions that the nilradical \(N\) of \(\Sigma\), i.e. the maximal connected nilpotent invariant subgroup of \(\Sigma\), is at least four-dimensional. Only the case of four-dimensional nilradicals has not been treated yet. The author proves that under the above assumptions the Lie algebra of \(N\) is isomorphic to the direct sum of a one-dimensional real Lie algebra and the three-dimensional nonabelian nilpotent real Lie algebra \(\langle x,y,z |[x,y] = z \rangle\). Models for planes admitting this type of collineation groups exist. Up to isomorphism there are precisely two four-dimensional nonabelian nilpotent real Lie algebras. One such Lie algebra is the one described above, the other Lie algebra, \({\math…
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51H10
zbMATH DE Number: 792561
zbMATH Keywords: translation plane
zbMATH Keywords: shift plane
zbMATH Keywords: nilradical
zbMATH Keywords: classification
zbMATH Keywords: 4-dimensional compact projective planes
zbMATH Keywords: duality
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 792561