MacLane method of group cohomologies and gauge theories on finite lattices
type: Item
instance of: scholarly article
title: MacLane method of group cohomologies and gauge theories on finite lattices
zbMATH Open document ID: 0822.20054
DOI: 10.1016/0034-4877(94)90018-3
author: T. Lulek
published in: Reports on Mathematical Physics
publication date: 1994-10-13T00:00:00Z
review text: It is well known that the set of inequivalent extensions of a group \(Q\) by an Abelian group \(T\), with given action of \(Q\) on \(T\), can be labelled by the elements of a second cohomology group. It is also well known that the classification of crystallographic space groups falls within this extension framework, as does the so-called magnetic translation group in the Brown-Zak theory of an electron moving in a crystal under the additional influence of a uniform magnetic field. A theoretical labelling is one thing, but the actual computation of the second cohomology group is not usually that easy. The MacLane free group method offers a considerable reduction in the combinatorial explosion inherent in any direct attack on the problem. This method is reviewed, as are the two applications referred to above.
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 81R99
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 639206
zbMATH Keywords: group extensions
zbMATH Keywords: computations of second cohomology groups
zbMATH Keywords: inequivalent extensions
zbMATH Keywords: crystallographic space groups
zbMATH Keywords: magnetic translation groups
zbMATH Keywords: MacLane free group method
reviewed by: N. B. Backhouse
MaRDI profile type: Publication
full work available at URL: 0034-4877(94)90018-3
OpenAlex ID: W2054212700
cites work: Cohomology theory in abstract groups. III. Operator homomorphisms of kernels
cites work: Q5583892
cites work: Generalized finite space groups
cites work: Q4023620
cites work: Anyons and quantum groups
cites work: Applications of \(S\)-functions to the quantum Hall effect and quantum dots
cites work: Magnetic translation groups as group extensions
cites work: Magnetic translation groups on a \(3\times 3\)-torus. Application of the Mac Lane method
cites work: The magnetic translation group in case \(N=2\)
cites work: Fiber structure of extensions of groups in crystallography
Recommended article: Quantum lattice gauge fields and groupoid \(\mathrm{C}^{\ast}\)-algebras
Recommended article: Q3750608
Recommended article: Homotopy classes of gauge fields and the lattice
Recommended article: Homotopy and duality in non-Abelian lattice gauge theory
Recommended article: Deformation quantization and homological reduction of a lattice gauge model
Recommended article: Loop gauge theory and group cohomology
Recommended article: Lie algebra cohomology and group structure of gauge theories
Recommended article: Lattice Wess-Zumino-Witten model and quantum groups
Recommended article: Gauge theories and Macdonald polynomials
Recommended article: Fermionic finite-group gauge theories and interacting symmetric/crystalline orders via cobordisms
description: scientific article; zbMATH DE number 639206