Left-invariant ${\rm G}_2^*$-structures of type III

TL;DR

This paper classifies certain special geometric structures called ${ m G}_2^*$-structures on 7D Lie groups, showing that only abelian holonomy algebras of dimension two or three occur for type III structures, with explicit examples provided.

Contribution

It identifies the possible abelian holonomy algebras of type III for left-invariant ${ m G}_2^*$-structures and constructs explicit Lie groups realizing these structures.

Findings

Only abelian subalgebras occur as holonomy algebras.

Holonomy algebras are of dimension two or three.

Explicit examples of Lie groups with these structures are given.

Abstract

We investigate left-invariant ${\rm G}_2^*$-structures on 7-dimensional Lie groups, focusing on those whose holonomy algebras are indecomposable and of type III, the latter meaning that the socle of the holonomy representation is maximal. Building on the classification of indecomposable holonomy algebras contained in $\mathfrak g_2^*$, we determine which ones arise as infinitesimal holonomy algebras of type III for left-invariant ${\rm G}_2^*$-structures. Our main result shows that only abelian subalgebras occur, and these are necessarily of dimension two or three. Moreover, we provide explicit Lie groups with left-invariant ${\rm G}_2^*$-structures realizing these abelian holonomies.