On the groups of isometries of simple para-Hermitian symmetric spaces
type: Item
instance of: scholarly article
title: On the groups of isometries of simple para-Hermitian symmetric spaces
zbMATH Open document ID: 1385.53042
author: Takuya Shimokawa
author: Kyoji Sugimoto
published in: Tsukuba Journal of Mathematics
publication date: 2017-11-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tkbjm/1506353558
review text: The goal of this paper is to completely determine the group of isometries of a simple para-Hermitian symmetric space. This allows to provide a complete description of the group of affine transformations of such a space with respect to its canonical affine connection. Indeed, it is shown that the group of isometries coincides with the group of affine transformations. The structure of \(I(G/H,g)/I(G/H,g)_0\) (the quotient group of the group of isometry over its connected component) is given in an extensive table. The authors start by recalling the definition and some known facts about para-Hermitian symmetric spaces (see for example [\textit{S. Kaneyuki} and \textit{M. Kozai}, Tokyo J. Math. 8, 81--98 (1985; Zbl 0585.53029)]). They then discuss the interrelationship between the group of isometries and the group \(\mathrm{Aut}(\mathfrak{g},\sigma_*)\) of Lie algebra automorphisms that commute with the involution \(\sigma_*\). This is done by giving a description of \(\mathrm{Aut}(\mathfrak{g},\sigma_*)\) and its correlation with the group of affine transformations leading to an isomorphism between these two and the equality between the group of isometries and the group of affine transformations. They conclude by explaining the construction of the table giving the structure of the quotient \(I(G/H,g)/I(G/H,g)_0\) and provide some examples of the computations (for the classical type namely \(\mathfrak{sl}(n,{\mathbb R})\) and \(\mathfrak{so}(n,n)\) and for the exceptional types namely \(\mathfrak{e}_{6(6)}\) and \(\mathfrak{e}_{7(-25)}\)).
reviewed by: Patrice Sawyer
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 6802083
zbMATH Keywords: para-Hermitian symmetric space
zbMATH Keywords: hyperbolic orbit type
zbMATH Keywords: absolutely simple Lie algebra
zbMATH Keywords: real form
zbMATH Keywords: group of isometries
MaRDI profile type: Publication
OpenAlex ID: W2760603517
DOI: 10.21099/TKBJM/1506353558
description: scientific article; zbMATH DE number 6802083