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Lifshitz symmetry: Lie algebras, spacetimes and particles

by José Figueroa-O'Farrill, Ross Grassie, Stefan Prohazka

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Submission summary

Authors (as Contributors): José Figueroa-O'Farrill · Ross Grassie · Stefan Prohazka
Submission information
Arxiv Link: (pdf)
Date accepted: 2022-11-21
Date submitted: 2022-10-06 12:43
Submitted by: Prohazka, Stefan
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We study and classify Lie algebras, homogeneous spacetimes and coadjoint orbits ("particles") of Lie groups generated by spatial rotations, temporal and spatial translations and an additional scalar generator. As a first step we classify Lie algebras of this type in arbitrary dimension. Among them is the prototypical Lifshitz algebra, which motivates this work and the name "Lifshitz Lie algebras". We classify homogeneous spacetimes of Lifshitz Lie groups. Depending on the interpretation of the additional scalar generator, these spacetimes fall into three classes: (1) ($d+2$)-dimensional Lifshitz spacetimes which have one additional holographic direction; (2) ($d+1$)-dimensional Lifshitz--Weyl spacetimes which can be seen as the boundary geometry of the spacetimes in (1) and where the scalar generator is interpreted as an anisotropic dilation; and (3) ($d+1$)-dimensional aristotelian spacetimes with one scalar charge, including exotic fracton-like symmetries that generalise multipole algebras. We also classify the possible central extensions of Lifshitz Lie algebras and we discuss the homogeneous symplectic manifolds of Lifshitz Lie groups in terms of coadjoint orbits.

Published as SciPost Phys. 14, 035 (2023)

Author comments upon resubmission

We thank the referees for their useful feedback and comments. We have taken it into consideration and we have made various changes as listed in the list of changes and the replies to the referees.

List of changes

Additionally to the changes that are described in the answers to the referees we have made the following change:

-) We corrected a mistake in L2 and L7 in Table 2. The changes in L7 required some explanation which we have added to Section 4.1.

Reports on this Submission

Anonymous Report 2 on 2022-10-26 (Invited Report)


The strengths are the same as in my previous report.

1). Very solid work.
2). Clearly written and presented.
3). Results are easy to look up due to summary of results and use of tables.


Also here nothing has changed. I my opinion there are no real weaknesses other than on the motivational front. It would have been nice with a bit more physical intuition or argumentation for why this classification is useful and how the results impact on certain physical problems.


I am happy with the changes made by the authors and I recommend the paper for publication.

  • validity: top
  • significance: good
  • originality: ok
  • clarity: top
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2022-10-7 (Invited Report)


SciPost acceptance criteria are fully met -- I recommend publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

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