Totally reducible holonomies of torsion-free affine connections

TL;DR

This paper classifies totally reducible holonomy Lie algebras of torsion-free affine connections, providing a comprehensive structure theorem that includes new examples and recovers some previously missing cases.

Contribution

It offers a complete classification of totally reducible holonomy Lie algebras for torsion-free connections, extending known results and introducing new constructions.

Findings

Provides a structure theorem for totally reducible holonomy Lie algebras.

Introduces new examples of non-irreducible holonomy algebras.

Recovers some irreducible holonomy algebras missing in previous classifications.

Abstract

That announcement gives the structure of totally reducible linear Lie algebras which are the Lie algebra of the holonomy group of (at least) one torsion-free connection. The result uses the (already known) classi cation of the irreducible ones and some previous (unpublished) works by the author giving the classi cation for the pseudo-riemannian totally reducible case. One describes those Lie subalgebras through a general structure theorem involving two constructions and some lists. These constructions give new examples of non irreducible totally reducible holonomy algebras and also recover some irreducible ones which seem missing in the previous classi cation.