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Type: Journal article
Title: Isometry Lie algebras of indefinite homogeneous spaces of finite volume
Author: Baues, O.
Globke, W.
Zeghib, A.
Citation: Proceedings of the London Mathematical Society, 2019; 119(4):1115-1148
Publisher: Wiley
Issue Date: 2019
ISSN: 0024-6115
Statement of
Oliver Baues, Wolfgang Globke and Abdelghani Zeghib
Abstract: Let be a real finite‐dimensional Lie algebra equipped with a symmetric bilinear form ⟨·,·⟩. We assume that ⟨·,·⟩ is nil‐invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomorphisms containing the adjoint representation of is an infinitesimal isometry for ⟨·,·⟩. Among these Lie algebras are the isometry Lie algebras of pseudo‐Riemannian manifolds of finite volume. We prove a strong invariance property for nil‐invariant symmetric bilinear forms, which states that the adjoint representations of the solvable radical and all simple subalgebras of non‐compact type of act by infinitesimal isometries for ⟨·,·⟩. Moreover, we study properties of the kernel of ⟨·,·⟩ and the totally isotropic ideals in in relation to the index of ⟨·,·⟩. Based on this, we derive a structure theorem and a classification for the isometry algebras of indefinite homogeneous spaces of finite volume with metric index at most 2. Examples show that the theory becomes significantly more complicated for index greater than 2. We apply our results to study simply connected pseudo‐Riemannian homogeneous spaces of finite volume.
Rights: © 2019 The Author(s). The Proceedings of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
RMID: 0030135028
DOI: 10.1112/plms.12252
Grant ID:
Appears in Collections:Mathematical Sciences publications

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