Mac Lane method in the investigation of magnetic translation groups

TL;DR

This paper explores the application of the Mac Lane method to analyze central extensions of the three-dimensional translation group by U(1), revealing connections to magnetic translation groups and limitations on factor systems.

Contribution

It demonstrates how the Mac Lane method relates to Zak's approach and identifies conditions under which factor systems can be realized for U(1) and its finite subgroups.

Findings

Factor systems by Zak and Brown are realizable only for U(1) and some finite subgroups.

The choice of generators as all nonzero vectors in T clarifies the relations between methods.

The study links algebraic group extensions to physical magnetic translation groups.

Abstract

Central extensions of the three-dimensional translation group T=Z^3 by the unitary group U(1) (a group of factors) are considered within the frame of the Mac~Lane method. All nonzero vectors t in T are considered to be generators of T. This choice leads to very illustrative relations between the Mac~Lane method and Zak's approach to magnetic translation groups. It is shown that factor systems introduced by Zak and Brown can be realized only for the unitary group U(1) and for some of its finite subgroups.