The novel class of collinear compensated magnets, dubbed altermagnets, has attracted immense research attention by the property of nonrelativistic spin splitting. More recently, the properties of altermagnets caused by relativistic spin-orbit coupling (SOC) became the topic of many investigations. The aim of the paper is to contribute to a deeper understanding of the formation of relativistic effects in altermagnets. The focus of the paper is on the phenomena of weak ferromagnetism (WFM) and relativistic modification of the electron states including the interplay of these phenomena. The main tools of the study are various types of relativistic calculations based on density functional theory combined with the symmetry analysis on the basis of spin space groups. The consideration is performed on two different levels. On the first level, the atomistic magnetic structure of weak ferromagnetic state is calculated and discussed. Both spin and orbital atomic moments are taken into account. We study the dependence of the WFM moment on the strength of the SOC and obtain a peculiar nonmonotonous type of dependence. We separate the contributions of the SOC of the metal and ligand atoms to WFM. A very interesting result is obtained in quasisymmetry (QS) calculation where only the component of the SOC collinear to the Néel vector is taken into account. In QS calculation the spin WFM is absent while the orbital WFM is present. This result reveals a principal difference in the formation of the spin and orbital magnetic moments. On the second level, the study is focused on the properties of individual electron states. We introduce the notion of the magnetic structure of the electron state (MSES). It is shown how the nonrelativistic collinear spin-MSES of both metal and ligand moments and nonrelativistic compensated orbital-MSES of the ligand moments transform into complex noncollinear three-dimensional MSESs of both spin and orbital nature giving large contributions to WFM. An important role in the formation of MSESs is played by the relativistic splitting of the accidental spin degeneracies at general k points filling the volume of the Brillouin zone. The formation of the regions of avoided crossings in the relativistic band structure is related to the nonmonotonous dependence of the WFM moment as the function of the SOC strength. The importance of the metal-ligand hybridization in the formation of the AM properties is discussed and illustrated by means of the comparison of the local density approximation (LDA) and $\mathrm{LDA}+U$ calculations. Most of the calculations are performed for MnTe, a prototypical material for the class of altermagnets.

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• Altermagnetism
• Electronic structure
• Magnetic order
• Spin-orbit coupling
• First-principles calculations