The Corepresentations of Continuous Groups

The theory of corepresentations of non-unitary groups G = G + a0G, where G denotes a unitary group and a0 is an antiunitary element, was formulated by Wigner [26], to whom belong the first applications of corepresentations in quantum mechanics. Space groups with antiunitary operations and their corepresentations subsequently found important applications in solid state physics. Lie groups with antilinear operations were not considered. Wigner’s theory of corepresentations was elaborated by a number of authors to the form of a powerful tool for investigations of physical properties of crystals and of magnetic crystals [1, 2, 3, 5, 20, 21]. It was applied in the investigations of symmetry changes at commensurate and incommensurate continuous magnetic phase transitions [8, 9, 10, 11], and to the problem of magnetocrystalline anisotropy of ferromagnetic crystals [14]. It was shown by Birman [1] that a non-unitary symmetry group can intervene in the classical description of a crystal in a state of thermodynamic equilibrium. The non-unitary group of the type G +KG, where G is a space group, and where K denotes the operation of complex conjugation, constitutes the complete symmetry group of the crystal-lattice dynamic problem. This group plays the basic role in establishing the one-to-one correspondence between vibration frequencies and irreducible corepresentations (coirreps). The application of the non-unitary group G + KG to a description of lattice vibrations elaborated in [1], opened the way for a further development in this field, made by Kovalev [15, 16, 17, 18, 21] and by Kovalev and Gorbanyuk [20]. These authors formulated another method of demonstrating that there exists the one-to-one correspondence between coirreps and frequencies of