On Lie algebra decompositions, related to spherical homogeneous spaces
type: Item
instance of: scholarly article
title: On Lie algebra decompositions, related to spherical homogeneous spaces
zbMATH Open document ID: 0797.17002
DOI: 10.1007/BF03026538
published in: Manuscripta Mathematica
publication date: 1994-10-13T00:00:00Z
full work available at URL: https://eudml.org/doc/155860
review text: Let \(G\) be a complex connected reductive group. An algebraic subgroup \(H\subset G\) is a spherical subgroup iff a Borel subgroup of \(G\) has an open orbit on \(G/H\). The author proves the following two theorems: (1) Let \(H_ 1\) and \(H_ 2\) be spherical subgroups of \(G\). Then there exists a torus \(A\) in \(G\) such that \[ L(G)= L(H_ 1)+ L(A)+ Ad(g).L(H_ 2) \] for all \(g\) in a Zariski open subset of \(G\) (\(L(G)\) is the Lie algebra of \(G\)). (2) Let \(H\) be a spherical subgroup of \(G\) and let \(s: G\to G\) be an antiholomorphic involution defining a real form \(G_ r\) of \(G\). Then there exists an \(s\)-invariant torus \(A\) in \(G\) such that \[ L(G)= L(G_ r)+ L(A)+ Ad(g).L(H) \] for all \(g\) in a Hausdorff open subset of \(G\).
Mathematics Subject Classification ID: 17B05
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 32M05
zbMATH DE Number: 492808
zbMATH Keywords: spherical homogeneous space
zbMATH Keywords: Lie algebra decomposition
zbMATH Keywords: reductive algebraic group
Wikidata QID: Q115390997
author: Dmitri Akhiezer
reviewed by: F. Pauer
MaRDI profile type: Publication
cites work: On the concept of the rank of a spherical homogeneous space
cites work: Quelques propriétés des espaces homogènes sphériques. (Some properties of spherical homogeneous spaces)
cites work: Valuations des espaces homogènes sphériques. (Valuations of spherical homogeneous spaces)
cites work: Sur les orbites fermées des groupes algébriques reductifs
cites work: Plongements d'espaces homogènes
cites work: Q5586507
cites work: Q4010540
cites work: Some Basic Theorems on Algebraic Groups
cites work: Q3993369
cites work: Complexity of action of reductive groups
cites work: The action of a real semisimple group on a complex flag manifold. I: Orbit structure and holomorphic arc components
description: scientific article; zbMATH DE number 492808