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Prolongations of Lie algebras and applications

Paul-Andi Nagy

TL;DR

The paper computes the skew-symmetric prolongation ${\Lambda^3V \cap (\Lambda^1V \otimes \mathfrak{g})}$ for orthogonal representations, establishing a dichotomy: for a proper irreducible $(\mathfrak{g},V)$ the prolongation vanishes unless $(\mathfrak{g},V)$ is the adjoint of a compact simple Lie algebra, in which case it is 1-dimensional and generated by the Cartan 3-form. This framework yields wide-ranging results: uniqueness of metric connections with totally skew-symmetric torsion, holonomy classifications for flat connections with 3-form torsion (Killing frames), a Plücker-type embedding program linked to $k$-Lie algebras, and a detailed study of metric $n$-Lie algebras. The authors also classify holonomy representations for connections with vectorial torsion, via invariant 4-forms and Casimir operators, and derive dimension-specific phenomena, notably a Spin(7)–related structure in dimension $8$. Altogether, the work blends Lie algebra prolongations, Berger algebras, and $G$-structure methods to advance geometric classification problems related to torsionful connections and their holonomy.

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 $Λ^1 \subseteq Λ^1 \otimes Λ^2$.

Prolongations of Lie algebras and applications

TL;DR

The paper computes the skew-symmetric prolongation for orthogonal representations, establishing a dichotomy: for a proper irreducible the prolongation vanishes unless is the adjoint of a compact simple Lie algebra, in which case it is 1-dimensional and generated by the Cartan 3-form. This framework yields wide-ranging results: uniqueness of metric connections with totally skew-symmetric torsion, holonomy classifications for flat connections with 3-form torsion (Killing frames), a Plücker-type embedding program linked to -Lie algebras, and a detailed study of metric -Lie algebras. The authors also classify holonomy representations for connections with vectorial torsion, via invariant 4-forms and Casimir operators, and derive dimension-specific phenomena, notably a Spin(7)–related structure in dimension . Altogether, the work blends Lie algebra prolongations, Berger algebras, and -structure methods to advance geometric classification problems related to torsionful connections and their holonomy.

Abstract

We study the skew-symmetric prolongation of a Lie subalgebra , in other words the intersection .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 .

Paper Structure

This paper contains 16 sections, 34 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Metric connections with torsion
  3. $3$-form torsion
  4. Vectorial torsion
  5. Prolongations of Lie algebras
  6. Algebraic curvature tensors and Berger algebras
  7. Structure results and linear holonomy algebras
  8. Reducible representations
  9. Uniqueness of connections with totally skew-symmetric torsion
  10. Killing frames on Riemannian manifolds
  11. A class of Plücker type embeddings and $k$-Lie algebras
  12. Metric $n$-Lie algebras
  13. The holonomy of connections with vectorial torsion
  14. Invariant forms and Casimir operators
  15. The classification
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(\mathfrak{g},V)$ be a faithful and irreducible representation of a Lie algebra onto an Euclidean space. If $\mathfrak{g}$ is proper then either:

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.1
  • ...and 54 more