CoChat
Sign Up
Cited by 92Noncommutative and Quantum Gravity TheoriesAdvanced Operator Algebra ResearchRelativity and Gravitational Theory

Quantum groups and noncommutative geometry

Open access
Majid, Shahn·2000·Journal of Mathematical Physics
DOI 10.1063/1.533331
Open this paper in CoChat — free accountOpen PDF

Abstract

Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalization of symmetry groups for certain integrable systems, and on the other as part of a generalization of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of space–time, in particular, amounts to a postulated new force or physical effect called cogravity.

Integrity signals

Registry

Crossref · 10.1063/1.533331

Cites

79 of 79 references verified0 citation errors · 0 not checkable

Statistics

Recomputing p-values needs the paper’s full text.

Claims the paper makes

We couldn’t retrieve this paper’s full text, so its claims and self-check weren’t run. The registry and citation checks still apply.

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The Hilbert space of a quantum tetrahedron is the sum over all irreducible representations assigned to its triangles of all invariant tensors (intertwiners).

    “T = j i Inv(j 0 ⊗ j 1 ⊗ j 2 ⊗ j 3 ) (10 )”

    From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [13]

  • For any group G, there is a category whose objects are non-degenerate spin networks and whose morphisms are non-degenerate spin foams.

    “It is important to note that, with these definitions of equivalence and composition we can define, for any given group G, a category F in which the objects are non-degenerate spin networks and the (non-degenerate) spin foams represent morphisms between them [13].”

    From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [13]

  • The commutation relations of the quantum operators show that the four triangle area operators commute with each other and with the parallelogram area operators, among which only two are independent, while the parallelogram area operators have nonvanishing commutators among themselves.

    “Analysing the commutation relations of the quantum operators corresponding to the triangle and parallelogram area operators, it turns out (see [13,58]) that, while the four triangle area operators commute with each other and with the parallelogram areas operators (among which only two are independent), the last ones have nonvanishing commutators among themselves.”

    From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [13]

  • The gluing procedure is consistent with the general spin foam formalism [13,14], which states that gluing two manifolds along a common boundary gives Z(M)Z(M')=Z(MM').

    “The gluing procedure used above is consistent with the formalism developed for general spin foams [13,14], and discussed in section 3, saying that, when we glue two manifolds M and M along a common boundary, the partition functions associated with them and with the composed manifold satisfy Z(M)Z(M ) = Z(MM )”

    From Spacetime geometry from algebra: spin foam models for non-perturbative quantum gravity (2001) · cites this paper as [13]

Cites

Flagged first, then by citations

Related

Authors
Majid, S.
Affiliations
University of London (GB)
Corresponding author
Majid, Shahn
ORCID
Majid, Shahn
Venue
Journal of Mathematical Physics
Publisher
AIP Publishing
Published
2000-06-01
Volume
41
Issue
6
Pages
3892-3942
ISSN
  • 0022-2488 (ISSN-L)
  • 1089-7658
DOI
10.1063/1.533331
OpenAlex
W2020855800
Cited by
92
References
79
Citations by year
2025: 2 · 2024: 5 · 2023: 4 · 2022: 1 · 2021: 6 · 2020: 3 · 2019: 1 · 2018: 3 · 2017: 3 · 2016: 2 · 2014: 2 · 2013: 3 · 2012: 6
FWCI
3.39
Type
journal article
Language
en
Open access
Locations
  • Journal of Mathematical Physics (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
Keywords
quantum groups, noncommutative geometry, noncommutative spacetime, integrable systems, symmetry groups, Planck-scale physics, quantum geometry

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review