Magnetic translation groups on a \(3\times 3\)-torus. Application of the Mac Lane method
type: Item
instance of: scholarly article
title: Magnetic translation groups on a \(3\times 3\)-torus. Application of the Mac Lane method
zbMATH Open document ID: 0818.20061
DOI: 10.1016/0034-4877(94)90020-5
author: D. Lipiński
published in: Reports on Mathematical Physics
publication date: 1995-08-17T00:00:00Z
review text: 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 \(\varepsilon=\exp (2\pi i/3)\).
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 82D25
zbMATH DE Number: 639208
zbMATH Keywords: group extensions
zbMATH Keywords: quantum mechanical groups
zbMATH Keywords: translation operators
zbMATH Keywords: planar finite lattices
zbMATH Keywords: periodic boundary conditions
zbMATH Keywords: gauge groups
zbMATH Keywords: inequivalent factor systems
zbMATH Keywords: magnetic translation groups
reviewed by: A. Janner
MaRDI profile type: Publication
full work available at URL: 0034-4877(94)90020-5
OpenAlex ID: W2092072485
cites work: The magnetic translation group in case \(N=2\)
cites work: Magnetic translation groups as group extensions
cites work: Cohomology theory in abstract groups. III. Operator homomorphisms of kernels
cites work: Q5583892
cites work: Q4745979
cites work: Q3241233
description: scientific article; zbMATH DE number 639208