Holonomy groups of $G_2^*$-manifolds

TL;DR

This paper classifies the possible holonomy algebras of manifolds with indecomposable torsion-free $G_2^*$-structures, expanding understanding of special geometric structures and their algebraic properties.

Contribution

It provides a classification of holonomy algebras for manifolds with indecomposable torsion-free $G_2^*$-structures and constructs examples on Lie groups.

Findings

Classification of holonomy algebras for $G_2^*$-manifolds

Realization of some holonomy algebras as Lie group metrics

Advancement in understanding of special geometric structures

Abstract

We classify the holonomy algebras of manifolds admitting an indecomposable torsion free $G_2^*$-structure, i.e. for which the holonomy representation does not leave invariant any proper non-degenerate subspace. We realize some of these Lie algebras as holonomy algebras of left-invariant metrics on Lie groups.