Admissible square quadruplets and semisimple symmetric spaces
type: Item
instance of: scholarly article
title: Admissible square quadruplets and semisimple symmetric spaces
zbMATH Open document ID: 1001.22011
published in: Advances in Mathematics
publication date: 2002-07-04T00:00:00Z
review text: Let \(\mathfrak g_{\mathbb C}\) be a complex semisimple Lie algebra and \(\mathfrak g\) be a real form of \(\mathfrak g_{\mathbb C}\). Suppose \(\sigma\) is an involution of \(\mathfrak g\). The \(1\) and \(-1\) eigenspaces of \(\sigma\) give the decomposition \(\mathfrak g=\mathfrak h +\mathfrak q\), \(\mathfrak g_{\mathbb C}=\mathfrak h_{\mathbb C} +\mathfrak q_{\mathbb C}\). The pair \((\mathfrak g, \mathfrak h)\) is then called a locally symmetric pair; if furthermore \(\mathfrak g\) is a compact real form of \(\mathfrak g_{\mathbb C}\) then \((\mathfrak g_{\mathbb C}, \mathfrak h_{\mathbb C})\) is called a complex locally symmetric pair. Now if \(\sigma\) and \(\theta\) are two commuting involutions of a complex semisimple Lie algebra \(\mathfrak g_{\mathbb C}\) with the corresponding decompositions \(\mathfrak g_{\mathbb C}=\mathfrak h_{\mathbb C} +\mathfrak q_{\mathbb C}=\mathfrak k_{\mathbb C} +\mathfrak p_{\mathbb C} \) then we have a quadruplet \((\mathfrak g_{\mathbb C}; \mathfrak h_{\mathbb C}, \mathfrak k_{\mathbb C}; \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C})\) with complex locally symmetric pairs \((\mathfrak h_{\mathbb C}, \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C})\) and \((\mathfrak k_{\mathbb C}, \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C})\). In the present paper the author proves the converse of this results, namely if \((\mathfrak h_{\mathbb C}, \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C})\) and \((\mathfrak k_{\mathbb C}, \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C}) \) are complex locally symmetric pairs then the involutions \(\sigma\) and \(\theta\) are commuting. Using these results the author describes the quadruplets \((\mathfrak g_{\mathbb C}; \mathfrak h_{\mathbb C}, \mathfrak k_{\mathbb C}; \mathfrak h_{\mathbb C}\cap \mathfrak k_{\mathbb C})\) corresponding to the non-Riemannian locally symmetric pair \((\mathfrak g, \mathfrak h)\).
reviewed by: Genkai Zhang
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1763524
zbMATH Keywords: Lie groups
zbMATH Keywords: Lie algebras
zbMATH Keywords: semisimple symmetric spaces
zbMATH Keywords: commuting involutions
author: Jing-Song Huang
MaRDI profile type: Publication
full work available at URL: aima.2001.2024
OpenAlex ID: W2041883702
cites work: Les espaces symétriques noncompacts
cites work: Q3897665
cites work: Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces
cites work: Q4947960
cites work: Lie groups beyond an introduction
cites work: Q3698971
cites work: Q3998356
DOI: 10.1006/AIMA.2001.2024
description: scientific article; zbMATH DE number 1763524