Saturated commutative spaces of Heisenberg type
type: Item
instance of: scholarly article
title: Saturated commutative spaces of Heisenberg type
zbMATH Open document ID: 1047.22021
published in: Acta Applicandae Mathematicae
publication date: 2004-08-20T00:00:00Z
review text: Let \(X=G/K\) be a connected Riemannian space with isometry group \(G\) and denote by \({\mathcal D}(X)^G\) the algebra of \(G\)-invariant differential operators. If \({\mathcal D}(X)^G\) is commutative then \(X\) is also called commutative and the pair \((G,K)\) is called a Gelfand pair. Several other characterizations of commutative spaces, for instance in terms of the Poisson bracket operation on the universal enveloping algebra, are discussed. \(X\) is said to be of Heisenberg type if in the Levi decomposition \(G=N\rtimes L\) one has \(L=K\). It is called saturated if \(N_F(K)^0=K\). A list of all saturated commutative spaces of Heisenberg type is given, for which \(\mathfrak{n}/\left[\mathfrak{n},\mathfrak{n}\right]\) is reducible.
reviewed by: Hubert Gollek
Mathematics Subject Classification ID: 22F30
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 2093668
zbMATH Keywords: Gel'fand pair
zbMATH Keywords: Lie groups
zbMATH Keywords: Poisson algebras
author: Oksana S. Yakimova
MaRDI profile type: Publication
full work available at URL: b:acap.0000024208.81031.d4
OpenAlex ID: W2042812256
DOI: 10.1023/B:ACAP.0000024208.81031.D4
description: scientific article; zbMATH DE number 2093668