
Journal of Lie Theory 23 (2013), No. 1, 001033 Copyright Heldermann Verlag 2013 SkewSymmetric Prolongations of Lie Algebras and Applications PaulAndi Nagy Dept. of Mathematics, University of Murcia, Campus de Espinardo, 30100 Espinardo  Murcia, Spain paulandi.nagy@um.es [Abstractpdf] \def\g{{\frak g}} \def\o{{\frak o}} \def\s{{\frak s}} We study the skewsymmetric prolongation of a Lie subalgebra $\g \subseteq \s\o(n)$, in other words the intersection $\Lambda^3 \cap (\Lambda^1 \otimes \g)$. We compute this space in full generality. Applications include uniqueness results for connections with skewsymmetric torsion and also the proof of the Euclidean version of a conjecture by FigueroaO'Farrill and Papadopoulos concerning a class of Pl\"uckertype embeddings. We also derive a classification of the metric kLie 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 $\Lambda^1 \subseteq \Lambda^1 \otimes \Lambda^2$. Keywords: Skewsymmetric prolongation, connection with skew symmetric, vectorial torsion. MSC: 53C05, 53C29 [ Fulltextpdf (385 KB)] for subscribers only. 