Cocalibrated structures on Lie algebras with a codimension one Abelian ideal

TL;DR

This paper classifies seven-dimensional Lie algebras with a codimension one Abelian ideal that admit cocalibrated G_2-structures, which are important for constructing special geometric structures via Hitchin's evolution equations.

Contribution

It provides a complete classification of such Lie algebras over real and complex fields admitting cocalibrated G_2-structures, advancing understanding of their geometric applications.

Findings

Classification of real Lie algebras with cocalibrated G_2-structures

Classification of complex Lie algebras with cocalibrated (G_2)_C-structures

Identification of structures suitable for Hitchin's evolution equations

Abstract

Cocalibrated G_2-structures and cocalibrated G_2^*-structures are the natural initial values for Hitchin's evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin_0(3,4), respectively. In this article, we classify which seven-dimensional real Lie algebras with a codimension one Abelian ideal admit such structures. Moreover, we classify the seven-dimensional complex Lie algebras with a codimension one Abelian ideal which admit cocalibrated (G_2)_C-structures.