The space of leftinvariant orthogonal almost complex structures on 6-dimensional Lie groups

TL;DR

This paper explores the structure of left-invariant orthogonal almost complex structures on 6-dimensional Lie groups, establishing an explicit isomorphism with complex projective space and providing formulas for their construction.

Contribution

It introduces an explicit isomorphism between the space of structures and P^3, and derives formulas for constructing these structures as compositions of rotations.

Findings

The space of structures is isomorphic to P^3.

Explicit formulas for structures as rotations are derived.

Provides a clear geometric understanding of these structures.

Abstract

The space $\mathcal{Z}$ of leftinvariant orthogonal almost complex structures, keeping the orientation, on 6-dimensional Lie groups is researched. To get explicit view of this space elements the isomorphism of $\mathcal{Z}$ and $\mathbb{C}P^3$ is used. The explicit formula for arbitrary leftinvariant orthogonal almost complex structure on 6-dimensional Lie group as composition of rotations is found.