Left invariant generalized complex and K\"ahler structures on simply connected four dimensional Lie groups: classification and invariant cohomologies

TL;DR

This paper classifies all left invariant generalized complex and K"ahler structures on four-dimensional simply connected Lie groups, and computes their invariant generalized cohomologies, providing a comprehensive understanding of these geometric structures.

Contribution

It provides a complete classification of these structures and explicit computations of their invariant generalized cohomologies on four-dimensional Lie groups.

Findings

Complete classification of generalized complex structures of type 1

Explicit calculations of generalized Dolbeault, Bott-Chern, and Aeppli cohomologies

Classification of invariant generalized K"ahler structures

Abstract

We give a complete classification of left invariant generalized complex structures of type 1 on four dimensional simply connected Lie groups and we compute for each class its invariant generalized Dolbeault cohomology, its invariant generalized Bott-Chern cohomology and its invariant generalized Aeppli cohomology. We classify also left invariant generalized K\"ahler structures on four dimensional simply connected Lie groups.