Skew-Symmetric Prolongations of Lie Algebras and Applications

We study the skew-symmetric prolongation of a Lie subalgebra g subset of so(n), in other words the intersection Lambda(3) boolean AND (Lambda(1) circle times 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 by Figueroa-O'Farrill and Papadopoulos concerning a class of Pluckertype embeddings. We also derive a classification of the metric k-Lie algebras (or Filipov algebras), in positive signature and finite dimension. 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 Lambda(1) subset of Lambda(1)circle times Lambda(2).