Abelian Balanced Hermitian structures on unimodular Lie algebras

TL;DR

This paper studies special Hermitian structures on unimodular Lie algebras, showing how their holonomy reduces and providing methods to construct and classify such structures, especially in 8-dimensional nilpotent cases.

Contribution

It characterizes unimodular Lie algebras with abelian balanced Hermitian structures and offers construction and classification techniques, notably in 8-dimensional nilpotent cases.

Findings

Holonomy reduces to a subgroup of SU(n-k).

Conditions for existence of these structures are identified.

Classification achieved for 8-dimensional nilpotent Lie algebras.

Abstract

Let $\mathfrak{g}$ be a $2n$-dimensional unimodular Lie algebra equipped with a Hermitian structure $(J,F)$ such that the complex structure $J$ is abelian and the fundamental form $F$ is balanced. We prove that the holonomy group of the associated Bismut connection reduces to a subgroup of $SU(n-k)$, being $2k$ the dimension of the center of $\mathfrak{g}$. We determine conditions that allow a unimodular Lie algebra to admit this particular type of structures. Moreover, we give methods to construct them in arbitrary dimensions and classify them if the Lie algebra is 8-dimensional and nilpotent.