Bounded bicircular domains in \({\mathbb{C}}^ n\)
type: Item
instance of: scholarly article
title: Bounded bicircular domains in \({\mathbb{C}}^ n\)
zbMATH Open document ID: 0715.32012
DOI: 10.1007/BF02568771
author: Dimitris-Panayotis Panou
published in: Manuscripta Mathematica
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155534
review text: It was proved by \textit{W. Kaup} and \textit{H. Upmeier} [Proc. Am. Math. Soc. 58, 129-133 (1976; Zbl 0337.32012)] that if D is the open unit ball in a complex Banach space E then there exists a \({\mathbb{C}}\)-linear subspace \(U\subset E\) such that the intersection \(A:=U\cap D\) equals the orbit G(0), where \(G=Aut(D)\) denotes the automorphism group of D. One can thus consider the following question: Given a bounded symmetric domain \(A\subset U\), where U is a linear subspace of \({\mathbb{C}}^ n\), which bounded circular domains \(D\subset {\mathbb{C}}^ n\) are extensions of A in the sense that \(A=C\cap U=Aut(D)(0)?\) The situation for extensions D of A which are Reinhardt domains was classified by \textit{T. Sunada} [Math. Ann. 235, 111-128 (1978; Zbl 0357.32001)]. The purpose of the present paper is to determine all extensions of a bounded symmetric domain \(A\subset U\) by bicircular domains \(D\subset E:={\mathbb{C}}^ n\). Here bicircular means that there is a linear subspace V with \(E=U\oplus V\) such that D is invariant under the bicircular group of transformations \((u,v)\mapsto (e^{is}u,e^{it}v)\) with s,t\(\in {\mathbb{R}}\). (Examples of such are provided by Reinhardt domains with respect to some direct decomposition of \({\mathbb{C}}^ n\) parallel to the coordinate axes.) In the case of bounded Reinhardt domains T. Sunada studied the Lie algebra \({\mathfrak g}\) of Aut(D) and showed that one has an analogue of the Cartan decomposition \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\). In this paper it is first proved that the Lie algebra of Aut(D) is reductive. Then the notion of hermitian representations of \(J^*\)-triples is introduced to describe the isomorphy classes of bicircular domains. A natural one-to-one correspondence between hermitian representations of \(J^*\)-triples and unitary representations of complex reductive Lie algebras which are the complexifications of the derivation algebras of the corresponding \(J^*\)-triples is construc…
reviewed by: Bruce Gilligan
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 22E60
zbMATH DE Number: 4177505
zbMATH Keywords: extensions
zbMATH Keywords: bounded symmetric domain
zbMATH Keywords: bicircular domains
zbMATH Keywords: bounded Reinhardt domains
zbMATH Keywords: hermitian representations
zbMATH Keywords: complex reductive Lie algebras
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4177505