2007/12/31 by Paul-Andi Nagy · 1 citation
Mathematics · #math.DG #msc:53C12 #msc:53C24 #msc:53C55
published as Journal of Lie Theory 23 (2013), No. 1, 1-33 · New version. Proofs shortened, one section added on flat connections with 3-form torsion
arxiv created 2008/06/05 · arxiv updated 2012/08/08
We study the skew-symmetric prolongation of a Lie subalgebra \g ⊆ \mathfrakso(n), in other words the intersection Λ3 ∩ (Λ1 ⊗ \g).We compute this space in full generality. Applications include uniqueness results for connections with skew-symmetric torsion and also the proof of the Euclidean version of a conjecture posed in \citeofarill concerning a class of Plücker-type embeddings. We also derive a classification of the metric k-Lie algebras (or Filipov algebras), in positive signature and finite dimension. Prolongations of Lie algebras can also be used to finish the classification, started in \citedatri, of manifolds admitting Killing frames, or equivalently flat connections with 3-form torsion. Next we study specific properties of invariant 4-forms of a given metric representation and apply these considerations to classify the holonomy representation of metric connections with vectorial torsion, that is with torsion contained in Λ1 ⊆ Λ1 ⊗ Λ2.