On harmonic tensors on three-dimensional Lie groups with left-invariant Riemannian metric

The three-dimensional Lie groups with left-invariant Riemannian metric and almost harmonic Schouten-Weyl tensor are discussed. The convolution of the Schouten-Weyl tensor is used in the direction of any vector that defines an antisymmetric 2-tensor and the structure of three-dimensional Lie groups and algebras with left-invariant Riemannian metric for which this tensor is harmonic. The Schouten-Weyl tensor is almost-harmonic if and only if the Lie algebra of the group is one of the types, the left-invariant Riemannian metric is homothetic to the standard metric, and the Schouten-Weyl tensor is trivial. The Schouten-Weyl tensor is almost harmonic if and only if the matrix of structure constants of the Lie algebra of G and the Schouten-Weyl tensor is trivial.