Magnetic translation groups on a \(3\times 3\)-torus. Application of the Mac Lane method
The author considers all inequivalent group extensions: \(0\to\mathbb{Z}_ 3\to G\to\mathbb{Z}_ 3\times\mathbb{Z}_ 3\to 1\), with trivial action of \(\mathbb{Z}_ 3\times\mathbb{Z}_ 3\) on \(\mathbb{Z}_ 3\). The aim is to study the problem of a quantum mechanical group of translation operators for an electron moving in a planar finite lattice (defined by corresponding Born-von Kármán periodic boundary conditions) with \(\mathbb{Z}_ 3\) playing the role of a gauge group. The determination of all inequivalent factor systems \(m\) is based on a method by Mac Lane for constructing the equivalence classes of group extensions \(0\to T\to G\to Q\to 1\) \((\Delta)\), for \(T\) abelian, \(Q\) finite and a given action \(\Delta: Q\to\Aut T\). For the case considered \((Q=\mathbb{Z}_ 3\times\mathbb{Z}_ 3\), \(T=\mathbb{Z}_ 3\) and \(\Delta\) = identity), the author finds 27 inequivalent factor systems \(m_ 1,\dots, m_{27}\) leading to three non-isomorphic groups \(G\), all of order 27. The first group, obtained for the trivial factor system \(m_ 1\), is \(\mathbb{Z}_ 3\times\mathbb{Z}_ 3\times\mathbb{Z}_ 3\). The second one, for \(m_ 2,\dots, m_{25}\), is \(\mathbb{Z}_ 3\times\mathbb{Z}_ 9\). The third one, the only non-abelian group, is given by the generators and relations: \(\langle R,S\mid R^ 3= S^ 3= (RS)^ 3= (R^{-1} S)^ 3 =1\rangle\). This last group is shown to be isomorphic to the magnetic translation group for the finite lattice \(\mathbb{Z}_ 3\times\mathbb{Z}_ 3\) generated by the translation operators \(\tau_ 1\), \(\tau_ 2\) satisfying the relation: \(\tau_ 1\tau_ 2=\tau_ 2\tau_ 1\varepsilon\) with \(\varepsilon=\exp (2\pi i/3)\).
scientific article; zbMATH DE number 639208