Algebraic polygons
type: Item
instance of: scholarly article
title: Algebraic polygons
zbMATH Open document ID: 0866.51005
author: Katrin Tent
author: Linus Kramer
published in: Journal of Algebra
publication date: 1997-07-07T00:00:00Z
review text: Let \({\mathfrak P}= ({\mathcal P}, {\mathcal L}, {\mathcal F})\) be a generalized \(n\)-gon \(n\geq 3\), and let \(d:{\mathcal P} \cup{\mathcal L} \to\{0, \dots,n\}\) be the distance function of \({\mathfrak P}\). Assume that \({\mathcal P}\) and \({\mathcal L}\) are \(K\)-varieties for some algebraically closed field \(K\). Then \({\mathfrak P}\) is called an algebraic polygon if the set \(D=\{(x,y)\in ({\mathcal P} \cup{\mathcal L}) \times ({\mathcal P} \cup{\mathcal L}) |d(x,y)= n-1\}\) is locally closed and \(f_{n-1}: D\to{\mathcal P} \cup {\mathcal L}\) is continuous, where \(f_{n-1} (x,y)=z\) if \(d(x,z)=n-2\) and \(d(y,z)=1\). The authors show that if \(K\) is of characteristic 0 then there are precisely 3 examples up to duality, namely the projective plane, the symplectic quadrangle, and the split Cayley hexagon over \(K\). The proof is achieved by using model theoretic concepts which allow the transfer of the problem from \(K\) to the field \(\mathbb{C}\) of complex numbers; in this particular case the theorem has already been proved by the first author in Math. Z. 223, No. 2, 333-341 (1996). As a corollary to this remarkable result one gets that the only irreducible spherical Tits systems with closed Borel subgroups in connected algebraic groups over algebraically closed fields of characteristic 0 are the classical ones.
reviewed by: Norbert Knarr
Mathematics Subject Classification ID: 51E24
Mathematics Subject Classification ID: 20E42
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 03C68
zbMATH DE Number: 922313
zbMATH Keywords: Lefschetz's principle
zbMATH Keywords: algebraic polygon
zbMATH Keywords: spherical Tits systems
MaRDI profile type: Publication
full work available at URL: jabr.1996.0179
OpenAlex ID: W4213363351
DOI: 10.1006/JABR.1996.0179
description: scientific article; zbMATH DE number 922313