Prolongations of Lie algebras and applications

TL;DR

This paper investigates the skew-symmetric prolongation of Lie subalgebras, providing full generality computations, applications to connections with skew-symmetric torsion, and classifications of metric k-Lie algebras and holonomy representations.

Contribution

It offers a comprehensive computation of skew-symmetric prolongations, applies results to torsion connections, and classifies metric k-Lie algebras and holonomy representations.

Findings

Computed the skew-symmetric prolongation space in full generality.

Proved uniqueness results for connections with skew-symmetric torsion.

Classified metric k-Lie algebras and holonomy representations.

Abstract

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. 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 $\Lambda^1 \subseteq \Lambda^1 \otimes \Lambda^2$.