The geometry of Jordan and Lie structures
type: Item
instance of: scholarly article
title: The geometry of Jordan and Lie structures
zbMATH Open document ID: 1014.17024
author: Wolfgang Bertram
published in: Lecture Notes in Mathematics
publication date: 2001-01-07T00:00:00Z
full work available at URL: t1754.htm
review text: In the last ten years the connection between geometry and Jordan algebras has been extended to a broader class of symmetric spaces. The objective of the book under review are general symmetric spaces, but also the potential applications of Jordan theory to harmonic analysis, from a geometric point of view. The method seems to be new. The present book tries to avoid the traditional approach of presentation by starting with geometric concepts (symmetric spaces, Lie groups) and adding additional geometric features (polarizations, almost complex structures). Thus, by applying the condition \(J^2 = \text{id}\) instead of \(J^2 = -\text{id}\) in the definition of complex structure, one arrives at paracomplex structures or polarizations, distinguished in ``straight'' and ``twisted'' case. Their formal definition is given in terms of the curvature tensor \(R\) and the invariant almost complex structure \(J\) of the complexified space. It is a main result of the book that every irreducible symmetric space of classical type admits a twisted complexification and that there are one, two or three inequivalent ones (mostly there exists only one). A surprising conclusion is that a simple Lie group is classical if and only if it admits a twisted complexification. These concepts lead to another type of classification of irreducible symmetric spaces. The conformal group (in the irreducible case) is characterized by a theorem that is a generalization of both, the classical Liouville theorem and the fundamental theorem of projective geometry. A Jordan version of the ``Campbell-Hausdorff formula'' is given globally for the symmetric space by means of algebraic conditions, the so called Jordan coordinates. The text is divided into two parts. The objective of the first one (Chapters I--V) is to develop an intrinsic theory of the Jordan-Lie functor and to describe its global geometric content. The theory presented in this part is easy and also general. The reader needs only ordinary tensor…
Mathematics Subject Classification ID: 17C36
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 17C37
Mathematics Subject Classification ID: 53-02
Mathematics Subject Classification ID: 17-02
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 17A40
Mathematics Subject Classification ID: 17C30
zbMATH DE Number: 1548753
zbMATH Keywords: Jordan algebras
zbMATH Keywords: symmetric spaces
zbMATH Keywords: paracomplex structures
zbMATH Keywords: polarizations
zbMATH Keywords: Jordan-Lie functor
zbMATH Keywords: Jordan triple systems
zbMATH Keywords: Jordan pairs
zbMATH Keywords: Kantor-Koecher-Tits construction
reviewed by: Demetra Demetropoulou-Psomopoulou
MaRDI profile type: Publication
OpenAlex ID: W1567419375
DOI: 10.1007/B76884
description: scientific article; zbMATH DE number 1548753