Isometry Lie algebras of indefinite homogeneous spaces of finite volume

摘要

Let g be a real finite-dimensional Lie algebra equipped with a symmetric bilinear form ⟨center dot,center dot⟩. We assume that ⟨center dot,center dot⟩ is nil-invariant. This means that every nilpotent operator in the smallest algebraic Lie subalgebra of endomorphisms containing the adjoint representation of g is an infinitesimal isometry for ⟨center dot,center dot⟩. 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 g act by infinitesimal isometries for ⟨center dot,center dot⟩. Moreover, we study properties of the kernel of ⟨center dot,center dot⟩ and the totally isotropic ideals in g in relation to the index of ⟨center dot,center dot⟩. 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.