A new class of kinematic spaces
type: Item
instance of: scholarly article
title: A new class of kinematic spaces
zbMATH Open document ID: 0965.51004
DOI: 10.1007/BF03322516
author: Yanyan Li
published in: Results in Mathematics
publication date: 2001-08-06T00:00:00Z
review text: The present paper is inspired by \textit{M. Marchi} and \textit{E. Zizioli} [Ann. Discrete Math. 18, 601-615 (1983; Zbl 0509.51011), cited as [MZ]] -- in which a large class of kinematic spaces (in the sense of H. Karzel) is introduced -- and the authors propose a generalization of the initial conditions stated therein. Indeed, in [MZ] one proves that given the kinematic space \((G,{\mathcal G},+)\) and given \(U\leq\Aut (G,{\mathcal G},+)\), then the semidirect product \(P=G\rtimes U\) becomes, in turn, a kinematic space provided the following condition holds \[ (K)\qquad \forall a\in G,\;\forall \mu\in U \setminus \{id\}\exists_1x\in G:x-\mu(x)=a. \] We notice that \((K)\) entails that the nontrivial automorphisms are fixed-point free (cf. condition K2 in [MZ]) and that the centre of \(P\) is trivial. The authors introduce the following (weaker) conditions: \[ K':\qquad \forall\mu,\nu\in U\setminus \{id\}: \text{Fix} \mu= \text{Fix} \nu, \] and \[ K'':\qquad \forall\mu\in U\setminus \{id\} \text{ the map }\mu-id\text{ is surjective} \] \((K''\) is just (K2) of [MZ]). These new conditions ensure that \(P=G\rtimes U\) is again a kinematic space and are indeed weaker than \((K)\): the authors exhibit an example involving an infinite dimensional vector space and a suitable subgroup \(U\leq GL(V,K)\); then \(P:=V \rtimes U\) verifies \((K')\) and \((K'')\), but not \((K)\). They also prove that \(P\) admits no semidirect product type splitting fulfilling \((K)\), since the centre of \(P\) is non-trivial. The paper is concise but clear and readable.
reviewed by: Elena Zizioli
Mathematics Subject Classification ID: 51J15
zbMATH DE Number: 1448546
zbMATH Keywords: kinematic spaces
zbMATH Keywords: semidirect product
MaRDI profile type: Publication
cites work: Q4357599
cites work: Q4771790
cites work: On a Class of Kinematic Spaces
full work available at URL: https://doi.org/10.1007/bf03322516
OpenAlex ID: W2017109062
description: scientific article; zbMATH DE number 1448546