Quasifields of symplectic translation planes
type: Item
instance of: scholarly article
title: Quasifields of symplectic translation planes
zbMATH Open document ID: 1179.51002
author: Norbert Knarr
published in: Journal of Combinatorial Theory. Series A
publication date: 2009-06-18T00:00:00Z
review text: Let \(V\) be a vector space over the field \(F\) and let \(\Sigma\) be a spread on \(V\), i.e.\ a covering of \(V\) by a collection of mutually complementary subspaces. If every element of \(\Sigma\) is totally isotropic w.r.t a non-degenerate symplectic bilinear form \(S : V\times V \rightarrow F\), then \(\Sigma\) is called a symplectic spread. Starting with a quasifield \(Q\) and a subfield \(F\) of the kernel \(K\) of \(Q\), the author shows that symplectic forms \(Q^2\times Q^2\rightarrow F\) turning \(\Sigma\) into a symplectic spread are in 1-1 correspondence to non-degenerate invariant bilinear forms \(B:Q\times Q\rightarrow F\) (i.e.\ \(B(xa,y)=B(x,ya)\) holds for all \(a,x,y\in Q\)) which in turn are in 1-1 correspondence to certain linear maps \(Q\rightarrow F\). If \(Q\) admits a non-degenerate invariant form, then the kernel \(K\) is commutative. Moreover, the right and the middle nucleus of \(Q\) coincide and both are contained in the center. From these results it follows that if a Desarguesian, Moufang or nearfield plane is symplectic, then the plane is actually Pappian.
reviewed by: Harald Löwe
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 12K05
zbMATH DE Number: 5566223
zbMATH Keywords: symplectic spread
zbMATH Keywords: quasifield
zbMATH Keywords: invariant bilinear form
zbMATH Keywords: kernel
zbMATH Keywords: nuclei
MaRDI profile type: Publication
full work available at URL: j.jcta.2008.11.012
OpenAlex ID: W2032232618
cites work: Q3433140
cites work: Commutative semifields and symplectic spreads.
cites work: Isomorphisms of symplectic planes
cites work: Q3891315
cites work: Simplectic spreads and finite semifields
cites work: Q4064611
DOI: 10.1016/J.JCTA.2008.11.012
description: scientific article; zbMATH DE number 5566223