It is the aim of this work to study product structures on four dimensional solvable Lie algebras. We determine all possible paracomplex structures and consider the case when one of the subalgebras is an ideal. These results are applied to the case of Manin triples and complex product structures. We also analyze the three dimensional subalgebras.
In this work we find necessary and sufficient conditions for a free nilpotent or a free metabelian nilpotent Lie algebra to be endowed with an ad-invariant metric. For such nilpotent Lie algebras admitting an ad-invariant metric the corresponding automorphisms groups are studied.Partially supported by Secyt-UNC and SCyT-UNR. Keywords: Free nilpotent Lie algebra, free metabelian nilpotent Lie algebra, ad-invariant metrics, automorphisms and derivations.MSC 2000: 17B01 17B40 17B05 17B30 22E25.
A family of naturally reductive pseudo-Riemannian spaces is constructed out
of the representations of Lie algebras with ad-invariant metrics. We exhibit
peculiar examples, study their geometry and characterize the corresponding
naturally reductive homogeneous structure.Comment: A shorter, clearer and more concise versio
We provide examples of naturally reductive pseudo-Riemannian spaces, in particular an example of a naturally reductive pseudo-Riemannian 2-step nilpotent Lie group (N, , N ), such that , N is invariant under a left action and for which the center is degenerate. The metric does not correspond to a bi-invariant one.
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