On exceptional Lie geometries

Schepper, Anneleen De; Schillewaert, Jeroen; Maldeghem, Hendrik Van; Victoor, Magali

doi:10.1017/fms.2020.57

Cited by 4 publications

(13 citation statements)

References 18 publications

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“…Indeed, one can then deduce that Ω has a residue which is (−1)lacunary, and these are listed in our current main result. Although it is not hard to predict the possibilities for Ω, it requires non-trivial arguments to actually prove this-this will be pursued in another paper, see [6]. The locally connected parapolar spaces we obtain are E 6,2 (K), E 7,1 (K), E 8,8 (K) (which are longroot geometries) and their relatives (more precisely: residues).…”

Section: Future Perspectivesmentioning

confidence: 90%

A Geometric Characterization of the Hjelmslev–Moufang Planes

Schepper

Schillewaert

Maldeghem

et al. 2021

The Quarterly Journal of Mathematics

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Hjelmslev–Moufang (HM) planes are point-line geometries related to the exceptional algebraic groups of type $\mathsf{E}_6$. More generally, point-line geometries related to spherical Tits buildings—Lie incidence geometries—are the prominent examples of parapolar spaces: axiomatically defined geometries consisting of points, lines and symplecta (structures isomorphic to polar spaces). In this paper we classify the parapolar spaces with a similar behaviour as the HM planes, in the sense that their symplecta never have a non-empty intersection. Under standard assumptions, we obtain that the only such parapolar spaces are exactly given by the HM planes and their close relatives (arising from taking certain restrictions). On the one hand, this work complements the algebraic approach to HM planes using Jordan algebras and due to Faulkner in his book ‘The Role of Nonassociative Algebra in Projective Geometry’, published by the American Mathematical Society in 2014; on the other hand, it provides a new tool for classification and characterization problems in the general theory of parapolar spaces.

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Section: Future Perspectivesmentioning

confidence: 90%

A Geometric Characterization of the Hjelmslev–Moufang Planes

Schepper

Schillewaert

Maldeghem

et al. 2021

The Quarterly Journal of Mathematics

Self Cite

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show abstract

“…In general, sheaf theory is used to recognize certain parapolar spaces from their common point residues. In [6] we have proved some local recognition theorems, and we will use those and one more, see Lemma 4.6. The proof of the latter resembles that of Lemma 5.1 of [6], but since there are some essential differences, we include it in full.…”

Section: Strategy Of the Proofmentioning

confidence: 99%

“…As for the second statement, we assume l < m (or, equivalently, 2d − 1 < n) and show that Ω is an admissible homomorphic image of a Grassmannian of a building. The proof roughly follows the strategy of the proof of Lemma 5.1 of [6]. It contains Lemma 5.2 of [6] (the proof of which was left to the reader) as a special case.…”

Section: Proof Of the Second Statement Of Conclusion (2) Of Theorem 41mentioning

confidence: 99%

“…The proof roughly follows the strategy of the proof of Lemma 5.1 of [6]. It contains Lemma 5.2 of [6] (the proof of which was left to the reader) as a special case.…”

Section: Proof Of the Second Statement Of Conclusion (2) Of Theorem 41mentioning

confidence: 99%

“…The Haircut Theorem uses a gap in the spectrum of dimensions of the singular subspaces of symplecta obtained by intersecting them with point-perps. Recently, three of the current authors worked on another type of characterisation of parapolar spaces, mainly of exceptional type, see [6]. Shult's Haircut Theorem would induce certain shortcuts in their arguments.…”

Section: Introductionmentioning

confidence: 99%

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