Prolongations of Lie algebras and applications

Abstract

We study the skew-symmetric prolongation of a Lie subalgebra $\g \subseteq \mathfrak{so}(n)$, in other words the intersection $Λ^3 \cap (Λ^1 \otimes \g)$.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 Plücker-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 ${\Lambda }^1\subseteq {\Lambda }^1\otimes {\Lambda }^2$.

New version. Proofs shortened, one section added on flat connections with 3-form torsion

References in corpus (3)

• On types of non-integrable geometries
• Conformal equivalence between certain geometries in dimension 6 and 7
• Flat nearly Kähler manifolds

Cited by in corpus (22)

• Three-Algebras and N=6 Chern-Simons Gauge Theories
• Constraining Maximally Supersymmetric Membrane Actions
• On the Lie-algebraic origin of metric 3-algebras
• Boundary Conditions for Interacting Membranes
• Brane Effective Actions, Kappa-Symmetry and Applications
• On the structure of k-Lie algebras
• Marginal Deformations and 3-Algebra Structures
• Truncated Nambu-Poisson Bracket and Entropy Formula for Multiple Membranes
• Lorentzian Lie n-algebras
• On Superspace Actions for Multiple M2-Branes, Metric 3-Algebras and their Classification
• M-theory backgrounds with 30 Killing spinors are maximally supersymmetric
• Metric Lie n-algebras and double extensions
• N=6 Membrane Worldvolume Superalgebra
• Three lectures on 3-algebras
• Compact supersymmetric solutions of the heterotic equations of motion in dimension 5
• On the affine group of a normal homogeneous manifold
• Irreducible complex skew-Berger algebras
• Classification of $(n+3)$-dimensional metric $n$-Lie algebras
• Special Riemannian geometries and the Magic Square of Lie algebras
• Multiple M-branes and 3-algebras
• A note on the uniqueness of the canonical connection of a naturally reductive space
• Metric 3-Leibniz algebras and M2-branes