Invariant complex structures on four-dimensional solvable real lie groups

Abstract

In this paper, invariant complex structures on four-dimensional, solvable, simply-connected real Lie groups are classified where the dimension of the commutator is less than three. The resulting complex surfaces corresponding to these structures are also determined. The classification problem is reduced to determining certain complex “structure” subalgebras of the complexifications of the four-dimensional, solvable real Lie algebras. Most of the eleven types of non-abelian solvable real Lie algebras do have complex structure subalgebras; three do not. Only three types of algebras have solvable complex structure subalgebras, and only one possesses both abelian and solvable complex structure subalgebras. Each of the possible homogeneous surfaces is represented in the list of resulting manifolds.