Spin factors as generalized Hermitian algebras
type: Item
instance of: scholarly article
title: Spin factors as generalized Hermitian algebras
zbMATH Open document ID: 1192.81193
author: David J. Foulis
author: Sylvia Pulmannová
published in: Foundations of Physics
publication date: 2009-07-10T00:00:00Z
review text: Let \(V\) be a real vector space with a symmetric positive definite bilinear form \(f:V\times V \to\mathbb R\). Positive-definite spin factor asociated with \((V,f)\) is the space \(R\oplus V\) equipped with the product \[ (\lambda+v)\circ (\mu+w)=\lambda\mu+f(v,w)+\lambda w+\mu v \] for all \(\lambda\), \(\mu\in\mathbb R\) and \(v,w\in V\). A Generalized Hermitian algebra (GH-algebra) is a subgroup \(G\) of the additive group of a ring \(R\) with the unit 1 such that \(G\) is a partially ordered abelian group with positive cone \(G^+\) satisfying certain requirements. A GH-algebra is of rank 2 if the projection structure of \(G\) is nontrivial and if every nontrivial projection is an atom. The paper analyzes the relationship between above mentioned structures. Main theorem states that positive-definite spin factors of dimension greater than one are mathematically equivalent to generalized Hermitian algebras of rank 2. For proving this result the authors derive many interesting properties of spin factors and GH-algebras.
reviewed by: Jan Hamhalter
Mathematics Subject Classification ID: 81R15
Mathematics Subject Classification ID: 81R25
Mathematics Subject Classification ID: 81P05
zbMATH DE Number: 5577215
zbMATH Keywords: spin factor
zbMATH Keywords: Jordan algebra
zbMATH Keywords: generalized Hermitian algebra
MaRDI profile type: Publication
full work available at URL: s10701-009-9279-8
OpenAlex ID: W2087910703
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DOI: 10.1007/S10701-009-9279-8
description: scientific article; zbMATH DE number 5577215