Skew-symmetric prolongations of lie algebras and applications

摘要

We study the skew-symmetric prolongation of a Lie subalgebra g ? so(n), in other words the intersection Λ 3 ∩ (Λ 1 ? 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 Plückertype 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 Λ 1 ? Λ 1 ? Λ 2.