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

O. P. Gladunova^a, E. D. Rodionov^b, V. V. Slavskii<sup>b

a</sup> Altai State University, Barnaul, Russia
^b Altai State Pedagogical Academy, Barnaul, Russia

Abstract: This paper studies three-dimensional Lie groups with left-invariant Lorentz metric and almost harmonic (i.e., having curl and divergence zero) Schouten–Weyl tensor. Moreover, by using the convolution of the Schouten-Weyl tensor in the direction of any vector, we define an antisymmetric $2$-tensor and study the structure of the three-dimensional Lie groups and algebras with left-invariant Lorentz metric for which this tensor is harmonic.

Keywords: Lie groups and algebras, left-invariant Lorentz metrics, harmonic tensor, Schouten–Weyl tensor.

UDC: 514.765

Received: 05.10.2008

 English version: Journal of Mathematical Sciences, 2014, 198:5, 505–545