Left-invariant CR structures on 3-dimensional Lie groups (Q2122090)

The aim of the paper is to clarify Cartan's classification results and approach on homogeneous CR \(3\)-manifolds in detail by using modern language. Every CR structure on a homogeneous CR \(3\)-manifolds is a left-invariant CR structure on a \(3\)-dimensional Lie group. The paper deals with the relation between the adjoint representation of the Lie group and the embedding of such CR structure in a complex \(2\)-dimensional manifold. Especially they emphasize on the fact that \(\operatorname{SL}_2(\mathbb{R})\) admits elliptic and hyperbolic families of left-invariant CR structures. They also present similar results for the Heisenberg and Euclidean groups.