Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces
type: Item
instance of: scholarly article
title: Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces
zbMATH Open document ID: 0685.22007
DOI: 10.1016/0001-8708(88)90066-7
author: Aloysius G. Helminck
published in: Advances in Mathematics
publication date: 1988-01-01T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/27141
review text: Let F be an algebraically closed field of char\(\neq 2\), and let G be a connected reductive algebraic group defined over F. In this paper the author classifies the isomorphism classes (under both inner and outer automorphisms) of (ordered) pairs (\(\theta\),\(\sigma)\) of commuting involutions of G. Given such a pair there exists a torus T of G which is maximally (\(\theta\),\(\sigma)\)-split: the classification goes via Araki type diagrams describing the simple roots of T, and their restrictions to the (\(\theta\),\(\sigma)\)-split part of T. If \(F={\mathbb{C}}\), and if G is semisimple, then given a pair (\(\theta\),\(\sigma)\) there exists a \(\theta\)- and \(\sigma\)-stable compact real form \({\mathfrak u}\) of G's Lie algebra \({\mathfrak g}\), with conjugation \(\tau\). Let \({\mathfrak g}_ 0\) be the real form of \({\mathfrak g}\) determined by the conjugation \(\theta\) \(\tau\). Then (\({\mathfrak g}_ 0,\sigma | {\mathfrak g}_ 0)\) is an infinitesimal semisimple symmetric pair; all such pairs are obtained in this way, and the above classification yields in particular a classification of semisimple symmetric pairs. The subclass of pairs (\(\theta\),\(\theta)\) gives rise to the Riemannian symmetric pairs in this way, and one obtains the old classification of Riemannian symmetric pairs by means of Araki diagrams. The classification of semisimple symmetric pairs for a real semisimple Lie algebra was achieved already by \textit{M. Berger} [Ann. Sci. Éc. Norm. Supér., III. Sér. 74, 85-177 (1957; Zbl 0093.356)], by means of a different method. The present method has the advantage that it clarifies much more of the fine structure of semisimple symmetric pairs.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 22E60
zbMATH DE Number: 4122249
zbMATH Keywords: connected reductive algebraic group
zbMATH Keywords: automorphisms
zbMATH Keywords: commuting involutions
zbMATH Keywords: torus
zbMATH Keywords: Araki type diagrams
zbMATH Keywords: simple roots
zbMATH Keywords: compact real form
zbMATH Keywords: Lie algebra
zbMATH Keywords: infinitesimal semisimple symmetric pair
zbMATH Keywords: semisimple symmetric pairs
zbMATH Keywords: Riemannian symmetric pairs
zbMATH Keywords: real semisimple Lie algebra
MaRDI profile type: Publication
OpenAlex ID: W1966708700
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description: scientific article; zbMATH DE number 4122249