Polarities of Schellhammer planes

Poppitz, Steffen; Stroppel, Markus

doi:10.1515/advgeom.2010.048

Cited by 3 publications

(1 citation statement)

References 20 publications

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“…One can use additional topological assumptions to replace the conditions on the order, cf. [11], [12], [13], [20], [21], [25], and [35]. Definition 1.2.…”

Section: Involutions Unitals and Weak Unitalsmentioning

confidence: 99%

Unitals over composition algebras

Knarr

Stroppel

2012

Forum Mathematicum

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We consider polar unitals in projective planes over composition algebras. We prove that the little projective group of such a unital is contained in the centralizer of the polarity defining the unital. This yields information about the full automorphism group of the unital; it is contained in the automorphism group of the normal subgroup generated by all translations of the unital. Over suitable ground fields (such as the real numbers) we obtain a complete description of the automorphism group of the unital in the sense that it coincides with the centralizer of the polarity. The action of a nilpotent regular normal subgroup is used to establish isomorphisms between unitals in spaces of different dimensions, and over different fields.

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“…One can use additional topological assumptions to replace the conditions on the order, cf. [11], [12], [13], [20], [21], [25], and [35]. Definition 1.2.…”

Section: Involutions Unitals and Weak Unitalsmentioning

confidence: 99%

Unitals over composition algebras

Knarr

Stroppel

2012

Forum Mathematicum

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Symmetric Quantum Sets and L-Algebras

Rump

2020

International Mathematics Research Notices

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Quantum analogues of sets are defined by two simple assumptions, allowing enumeration, reminiscent of the Gram–Schmidt orthogonalization process. It is shown that any symmetric quantum set is a classical set of irreducible components, and that each irreducible component of size $>3$ is representable by an orthomodular space over a skew field with involution. For finite or sufficiently large irreducible components, invariance of quantum cardinality is proved. Topological quantum sets are introduced as quantum analogues of topological spaces; irreducible ones of size $>3$ are shown to be representable by Hilbert spaces over ${\mathbb{R}}$, ${\mathbb{C}}$, or ${\mathbb{H}}$. Symmetric quantum sets are characterized as a class of $L$-algebras with an intrinsic geometry, and they are shown to be equivalent to Piron’s quantum formalism. Equivalences between symmetric quantum sets and several other structures are established. To any symmetric quantum set, a group with a right invariant lattice structure is associated as a complete invariant. A simple and self-contained proof of Solèr’s theorem is included, which is used to prove that sufficiently large irreducible symmetric quantum sets come from a classical Hilbert space.

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Substituting compact disks in stable planes

Löwen

2011

Advg

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Given two stable planes of dimension 2l containing compact topological 2l-disks whose boundaries are isomorphic unitals, we show that it is possible to substitute one of these disks with the other, thus changing the line system in one of the planes, and that this produces another stable plane. The proof uses intersection theory in singular homology in order to show that existence of interior intersection points of two secants can be detected by looking at their traces in the unital. We conclude with a class of concrete examples based on P. Sperner's 4-dimensional affine planes with group .

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Polarities of Schellhammer planes

Cited by 3 publications

References 20 publications

Unitals over composition algebras

Unitals over composition algebras

Symmetric Quantum Sets and L-Algebras

Substituting compact disks in stable planes

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