Killing tensors as irreducible representations of the general linear group
type: Item
instance of: scholarly article
title: Killing tensors as irreducible representations of the general linear group
zbMATH Open document ID: 1086.53071
author: Raymond G. Mclenaghan
author: Robert Milson
author: Roman G. Smirnov
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2005-02-22T00:00:00Z
review text: Let \((M^n, g_{\alpha,\beta})\) be an \(n\)-dimensional pseudo-Riemannian manifold of constant curvature. A Killing tensor of valence \(p\) is a symmetric tensor field \(h_{\alpha_1\dots\alpha_p}\) satisfying \(\nabla_{(\alpha_0}h_{\alpha_1\dots\alpha_p)}=0\). Let \({\mathcal K}^p\) denote the vector space of such tensor fields on \(M^n\). \textit{M.~Takeuchi} showed [Tsukuba J. Math. 7, 233--255 (1983; Zbl 0567.53017)] that \({\mathcal K}^p\) is isomorphic to a certain representation of the general linear group by invoking the Bott-Borel-Weil theorem. The authors construct an elementary isomorphism between \({\mathcal K}^p\) and this irreducible representation. This isomorphism is equivariant in the sense that the natural action of the isometry group corresponds to the restriction of the linear action to the appropriate subgroup.
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 2136384
zbMATH Keywords: Killing tensors
zbMATH Keywords: pseudo-Riemannian manifolds
zbMATH Keywords: manifolds of constant curvature
zbMATH Keywords: representations of the general linear group
reviewed by: Sergei S. Platonov
MaRDI profile type: Publication
full work available at URL: j.crma.2004.07.017
OpenAlex ID: W2021000789
cites work: Homogeneous vector bundles
cites work: Q4002278
cites work: Differential geometry, Lie groups, and symmetric spaces.
cites work: An extension of the classical theory of algebraic invariants to pseudo-Riemannian geometry and Hamiltonian mechanics
cites work: Killing tensor fields on spaces of constant curvature
cites work: Killing tensors in spaces of constant curvature
DOI: 10.1016/J.CRMA.2004.07.017
description: scientific article; zbMATH DE number 2136384