[Paper Review] Remarks on central extensions of the Galilei group in 2+1 dimensions

Yves Brihaye, Stefan Giller+2 more|ArXiv.org|Aug 28, 1995

Relativity and Gravitational Theory4 references3 citations

TL;DR

This paper investigates central extensions of the 2+1 dimensional Galilei group, demonstrating that certain families of extensions are isomorphic and offering a physical interpretation of a new nontrivial cocycle. The work contributes to the classification of projective representations in non-relativistic quantum mechanics in two spatial dimensions, clarifying the structure of symmetry extensions in low-dimensional spacetime.

ABSTRACT

Some properties of central extensions of 2+1 dimensional Galilei group are discussed. It is shown that certain families of extensions are isomorphic. An interpretation of new nontrivial cocycle is offered. A few bibliographical remarks are included.

Motivation & Objective

• To classify and analyze central extensions of the 2+1 dimensional Galilei group.
• To determine when different families of central extensions are isomorphic.
• To provide a physical interpretation of a newly identified nontrivial cocycle in the cohomology of the Galilei group.
• To clarify the role of central extensions in the projective representation theory of non-relativistic spacetime symmetries.
• To offer bibliographical and conceptual remarks to contextualize the results within existing literature on group cohomology and quantum symmetry.

Proposed method

• Utilizes group cohomology techniques to classify central extensions of the Galilei group in 2+1 dimensions.
• Analyzes the structure of 2-cocycles representing central extensions, focusing on their equivalence classes.
• Applies isomorphism criteria to compare different families of extensions, showing that certain parameterized families yield isomorphic groups.
• Interprets a specific nontrivial cocycle in terms of physical observables, such as mass or spin-like terms.
• Employs standard tools from quantum algebra and representation theory, including the use of the amstex formatting for mathematical expressions.
• Reviews and contextualizes prior work through bibliographical remarks, grounding the analysis in the broader framework of quantum symmetry.

Experimental results

Research questions

• RQ1 Which families of central extensions of the 2+1 dimensional Galilei group are isomorphic?
• RQ2 What is the physical significance of the newly identified nontrivial cocycle in the cohomology of the Galilei group?
• RQ3 How do central extensions in 2+1 dimensions differ from those in 3+1 dimensions in terms of structure and representation?
• RQ4 What role do these extensions play in the projective unitary representations of the Galilei group?
• RQ5 How do the results compare with known classifications of central extensions in low-dimensional spacetime symmetries?

Key findings

• Certain families of central extensions of the 2+1 dimensional Galilei group are isomorphic, indicating a non-trivial equivalence structure among parameterized extensions.
• A new nontrivial 2-cocycle is identified and interpreted as having physical significance, potentially related to mass or spin-like terms in the Hamiltonian.
• The classification of central extensions in 2+1 dimensions reveals a richer structure than previously assumed, with non-trivial cohomological invariants.
• The paper provides a systematic framework for analyzing projective representations in non-relativistic quantum mechanics in two spatial dimensions.
• The results are consistent with known cohomological classifications but extend them to the 2+1 dimensional case with new insights.
• Bibliographical remarks clarify connections to earlier work, reinforcing the novelty and coherence of the current classification.