A uniform description of compact symmetric spaces as Grassmannians using the magic square
type: Item
instance of: scholarly article
title: A uniform description of compact symmetric spaces as Grassmannians using the magic square
zbMATH Open document ID: 1280.53050
author: Yongdong Huang
author: Naichung Conan Leung
published in: Mathematische Annalen
publication date: 2011-05-23T00:00:00Z
review text: The authors derive a representation for symmetric spaces of the compact type which extends the ``magic square'' representation of compact semisimple Lie groups due to Freudenthal and Tits. In the classical magic square, the Lie algebras of compact semisimple Lie groups are described by means of \((\mathbb A\otimes\mathbb B)^n\), where \(\mathbb A\) and \(\mathbb B\) are normed division algebras. The authors use the notation \(\{ P \subset V \}\) to denote the Grassmannian of planes of type \(P\) (given by dimension and certain metric conditions) in the vector space \(V\). For example, \(\{\mathbb R^k\subset\mathbb R^n\}\) denotes the Grassmannian \(O(n)/O(k)\times O(n-k)\) of \(k\)-planes in \(\mathbb R^n\), and \(\{\mathbb R^n\subset\mathbb C^n\}\) denotes the Grassmannian \(U(n)/O(n)\) of real Lagrangian subspaces in \(\mathbb C^n\). For a division algebra \(\mathbb A=\mathbb C,\mathbb H,\mathbb O\) let \(\frac{\mathbb A}{2}\) denote \(\mathbb R,\mathbb C,\mathbb H\), respectively. With this notation the main result of the paper is a description of symmetric spaces of compact type as one of the following types of Grassmannians: {\parindent=0.7cm\begin{itemize}\item[(I)]Isotropic Grassmannians: \(\{(\mathbb A\otimes\mathbb B)^n\subset (\mathbb A\otimes\mathbb B)^n\oplus(\mathbb A\otimes\mathbb B)^n\}^{\sigma}\), where \(\sigma\) is an involution for an indefinite symmetric bilinear form of neutral signature.\item[(II)] Grassmannians: \(\{(\mathbb A\otimes\mathbb B)^k\subset (\mathbb A\otimes\mathbb B)^n\}\).\item[(III)] Lagrangian Grassmannians: \(\{(\frac{\mathbb A}{2}\otimes\mathbb B)^n\subset(\mathbb A\otimes\mathbb B)^n\}\). \item[(IV)] Double Lagrangian Grassmannians: \(\{(\frac{\mathbb A}{2}\otimes\frac{\mathbb B}{2})^n\oplus(j_1\frac{\mathbb A}{2}\otimes j_2\frac{\mathbb B}{2})^n \subset(\mathbb A\otimes\mathbb B)^n\}\), where \(j_1\), \(j_2\) are one of \(i\in\mathbb C\), \(j\in\mathbb H\) or \(e^3\in\mathbb O\). \end{itemize}}
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 17A35
zbMATH DE Number: 5898900
zbMATH Keywords: compact symmetric spaces
zbMATH Keywords: magic square
zbMATH Keywords: division algebras
MaRDI profile type: Publication
full work available at URL: s00208-010-0549-8
OpenAlex ID: W2034875135
Wikidata QID: Q29040063
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DOI: 10.1007/S00208-010-0549-8
description: scientific article; zbMATH DE number 5898900