Two-step solvable SKT shears

TL;DR

This paper classifies two-step solvable Lie groups with SKT structures using shear construction, providing new insights into their structure and extending the classification to specific algebra cases, especially in dimension six.

Contribution

It introduces a shear-based method to classify two-step solvable SKT Lie groups, covering various algebraic cases and advancing the understanding of their structure.

Findings

Classification of two-step solvable SKT Lie groups using shear data.

Detailed structure results for specific algebra cases.

Partial classification of six-dimensional SKT algebras.

Abstract

We use the shear construction to construct and classify a wide range of two-step solvable Lie groups admitting a left-invariant SKT structure. We reduce this to a specification of SKT shear data on Abelian Lie algebras, and which then is studied more deeply in different cases. We obtain classifications and structure results for $\mathfrak{g}$ almost Abelian, for derived algebra $\mathfrak{g}'$ of codimension 2 and not $J$-invariant, for $\mathfrak{g}'$ totally real, and for $\mathfrak{g}'$ of dimension at most 2. This leads to a large part of the full classification for two-step solvable SKT algebras of dimension six.