Prolongations of Lie algebras and applications

Understand

We study the skew-symmetric prolongation of a Lie subalgebra $\g \subseteq \mathfrak{so}(n)$, 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 skew-symmetric torsion and also the proof of the Euclidean version of a conjecture posed in \cite{ofarill} concerning a class of Pl\"ucker-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 \cite{datri}, of manifolds admitting Killing frames, or equivalently flat connections with 3-form torsion.