Enveloping algebra valued gauge transformations for non-abelian gauge groups on non-commutative spaces

Branislav Jurco; Julius Wess; Peter Schupp; Stefan Schraml

REVIEW

Enveloping algebra valued gauge transformations for non-abelian gauge groups on non-commutative spaces

Branislav Jurco, Stefan Schraml, Peter Schupp, Julius Wess

High Energy Physics - Theory · Submitted 30 Jun 2000 · 18 references

Pith's verdict: The first-order non-abelian Seiberg-Witten construction is explicit and correct; the abstract's finite-field-content claim is only established to first order in theta, with the all-orders step delegated to a paper listed as in preparation.

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arXiv IDhep-th/0006246 (version 1)

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CitationBranislav Jurco, Julius Wess, Peter Schupp, Stefan Schraml. "Enveloping algebra valued gauge transformations for non-abelian gauge groups on non-commutative spaces." arXiv:hep-th/0006246 (2000). https://pith.science/paper/OZJTIYVH

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Worth your time, with one caveat about scope. The real content here is a concrete, checkable first-order construction: an enveloping-algebra-valued gauge parameter and connection expressed entirely in terms of the Lie-algebra parameter alpha1 and the Lie-algebra field a1, Eqs. (3.22)-(3.23). That is the thing people actually use, and it is derived, not asserted. The closure check in (3.29)-(3.35) is a direct star-product computation and I could not break it. Section 4 is a nice bonus: an integral with the trace property and a Stokes theorem for the canonical case, which is what lets you write a gauge-invariant action at all. The field strength in (3.40) matches what you would get from Seiberg-Witten directly, which is a good consistency sign. So the paper does what its title says at first order, cleanly, and the algebra holds up under checking. Credit where due: this is the paper that made finite-field-content noncommutative Yang-Mills look computable rather than formal, and it did so with formulas you can actually put into a calculation. Now the soft spot, and it is real but narrower than it looks. The abstract's claim [abstract:s1] that a finite number of components suffices is proven only to first order in theta. The all-orders step rests on the grading assumption stated in Section 3, that alpha_n and a_n are of order theta^{n-1} and degree n in the generators. That is asserted, not derived. The paper then says existence follows from the general setting of the previous chapter, which is a mispointer: Section 2 sets up the star product and proves nothing about existence. The honest pointer is reference [6], listed as in preparation. Bare existence of the map is fine, since [4] is published and [5] was in press, but the specific grading that makes the finite-content claim work is not established here. The authors do acknowledge that closure is shown only to first order, so this is a scope limitation rather than a hidden gap. Secondary: heavy self-citation to [1], [4], [5], [6] with overlapping authors, though the cited results are published or in press, so it is not a citation problem so much as a sign of how tightly this group owns the program. Who should read it: anyone building noncommutative gauge models who needs the explicit first-order map and the action prescription. Send it out. The first-order results deserve referee attention and the all-orders question is exactly what a referee should press on.

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Major objections (2)

[§3, grading assumption before Eq. (3.16)] The sentence 'We always assume that α_n and a_n^ρ are of power θ^{n−1} and polynomials in t of degree n' is the condition that makes the θ-expansion and the enveloping-algebra expansion line up term by term, and it is asserted rather than derived. The paper then writes that existence to all orders 'follows from the existence of the Seiberg-Witten map in the general setting of the previous chapter,' but §2 only sets up the ∗-product formalism and does not prove general existence; the explicit all-orders non-abelian construction is delegated to reference [6], listed as 'in preperation.' Since the abstract's general claim [abstract:s1] depends on this grading, the paper should either sketch why each θ-derivative in the ⊛-product brings one extra power of t (so that the two expansions align) or explicitly flag the all-orders statement as conditional on [6].
[§4, Eq. (4.20)] The vanishing of δW requires Tr([α∗, L]) = 0, which in turn requires the trace over the enveloping algebra to be cyclic with respect to the full ∗-product, including the t-part. The paper shows the x-part cyclicity in Eqs. (4.11)-(4.16) and states that 'the trace has to be taken for the group generators,' but it does not state or show the t-part cyclicity. Since the gauge invariance of the action is the paper's stated route to a dynamics, this should be made explicit, even if only by a short argument that the ⊛-product trace is cyclic by construction.

Minor objections (8)

[Eq. (3.14)] Eq. (3.14) has the opposite sign to Eq. (3.16) for the θ-linear term; the sign in Eq. (3.16) is the one that follows from the star product (3.3), so Eq. (3.14) appears to contain a sign typo.
[Eq. (4.9)] Eq. (4.9) is incomplete: the displayed chain ends with '= .' and should be finished.
[Eq. (4.2)] Eq. (4.2) has index inconsistencies (d_{μσρκ}d_{νασβ} versus d_{νσρκ}d_{μασρ}); the indices should be checked and made uniform.
[Title page and body] There are several typographical errors that should be corrected: 'Instiut' should be 'Institut', 'Vivatgasse' should be 'Vivatsgasse', 'explicitely' should be 'explicitly', 'preperation' should be 'preparation', 'choosen' should be 'chosen', 'totaly' should be 'totally', 'derivates' should be 'derivatives', 'Procee ding' should be 'Proceeding', 'whithin' should be 'within', and 'aswell' should be 'as well'.
[References] Reference [6] is listed as 'in preperation' and should be updated if it has appeared; reference [1] is listed as 'in press' and should be updated to the published version.
[Abstract] The abstract says 'An enveloping algebra valued gauge field is constructed,' but the paper actually constructs the gauge parameter and connection explicitly only to first order in θ; the abstract could be more precise about the order at which the construction is demonstrated.
[Title] The title refers to 'non-abelian gauge groups,' but the explicit construction is for a general Lie algebra and the paper does not discuss the abelian case or the U(1) limit; a brief remark on this would help the reader.
[§4, Eq. (4.6)] The claim that derivatives do not have to be introduced separately for the canonical structure, via the relation x̂^α − iθ^{αρ}∂̂_ρ = 0, is a known construction (see references [17,18]), but the paper should clarify that dividing the algebra by the ideal generated by this relation is a choice, not a necessity.

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The paper

The paper claims

Noncommutative gauge theory can be rewritten with only the ordinary number of gauge fields, because every higher algebraic component is fixed by the Seiberg-Witten map.

The reading

On a noncommutative space, an infinitesimal gauge transformation does not stay inside the Lie algebra: commutators and anticommutators of the generators pile up higher and higher powers, so the natural home for both the transformation parameter and the connection is the infinite-dimensional enveloping algebra. That looks fatal, since it would require infinitely many coordinate-dependent parameters and infinitely many gauge fields. The paper argues that this tower is not free: if the parameter and the connection are built with the Seiberg-Witten map, each higher component is a function of the ordinary Lie-algebra-valued parameter, the ordinary Lie-algebra-valued gauge field, and their derivatives. The payoff claimed in the abstract is that one can then write a dynamics for a finite number of gauge field components on a noncommutative space, with the same field content as an ordinary gauge theory. The paper verifies this explicitly to first order in the noncommutativity parameter $[abstract:s0]$ and argues the general order by matching powers of $[abstract:s1]$.

What carries the argument

The load-bearing device is a single star product on one big algebra whose generators are the noncommutative coordinates together with the Lie algebra generators, so that $[abstract:s0]$ the space and the internal symmetry are treated on the same footing. Gauge fields enter as covariant coordinates $[abstract:s0]$, whose transformation law forces the connection to be enveloping-algebra-valued. The Seiberg-Witten map is then the mechanism that ties every higher power of the generators to the Lie-algebra data, and the crucial structural assumption is the grading: the term of order $[abstract:s0]$ in the noncommutativity parameter is taken to be a polynomial of degree $n$ in the generators, which is what makes the two expansions line up order by order. Because higher generator powers correspond to higher powers of the structure constants, this is simultaneously an expansion in the gauge coupling. The paper also shows that for the canonical structure the derivatives need not be introduced separately, since $[abstract:s0]$ commutes with all coordinates and can be set to zero, and that an integral with the cyclic and Stokes properties exists, which is what allows a gauge-invariant action.

Extended reading notes

Core claim

The paper's central claim is that an enveloping-algebra-valued gauge parameter $[abstract:s0]$ and an enveloping-algebra-valued connection on a canonically noncommutative space can both be restricted so that their components are determined by a Lie-algebra-valued parameter $[abstract:s0]$ and a Lie-algebra-valued gauge field through the Seiberg-Witten map. Concretely, writing the expansion in powers of the noncommutativity parameter $[abstract:s0]$, the terms of order $[abstract:s0]$ are assumed to be polynomials of degree $n$ in the group generators, and the paper shows that this grading is consistent: the variation of the connection closes on the same restricted form, and the composition of two restricted transformations reproduces the commutator of the underlying Lie-algebra transformations. To first order the solution is written out explicitly, with the parameter and connection corrections expressed through the ordinary field strength and its derivatives, and the resulting field strength transforms covariantly under the restricted transformation. The authors then state that the same counting holds to all orders, so that a gauge-invariant action can be built from a finite number of components $[abstract:s1]$.

Load-bearing premise

The step from the explicit first-order calculation to a claim about all orders rests on an assumed bookkeeping rule, stated in Section 3 rather than proved there: the correction at order $[abstract:s0]$ in the noncommutativity parameter is taken to be a polynomial of degree $n$ in the group generators, and the all-orders statement is then delegated to a paper listed as in preparation.

Editorial extensions

If this is right

A noncommutative gauge theory can be written with exactly as many independent gauge fields and transformation parameters as the corresponding commutative Lie-algebra gauge theory.
A gauge-invariant action can be built from the covariant field strength $[abstract:s0]$ using a trace over the group generators and an integral with the cyclic property, and it reduces to the usual action as the noncommutativity parameter goes to zero.
The restricted transformations close: the commutator of two restricted transformations is again a restricted transformation, so the finite-parameter truncation is consistent rather than a first-order accident.
The construction extends from the canonical case to Lie-type and quantum-plane noncommutativity with coordinate-dependent $[abstract:s0]$, with the same first-order formulas written in terms of the appropriate $[abstract:s0]$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the grading holds at every order, noncommutative corrections to a nonabelian gauge theory cannot introduce new propagating degrees of freedom; all novelty would have to sit in the derivative structure of the star product, not in extra fields.
The grading assumption is directly testable at second order: compute the order-$[abstract:s0]$ terms for a simple group such as $[abstract:s0]$ and check whether the corrections are indeed degree-two polynomials in the generators, or whether terms of other degrees appear.
Because the generator degree tracks powers of the coupling, the map as constructed is a simultaneous expansion in the noncommutativity parameter and in the gauge coupling; the regime where it is trustworthy is therefore narrower than a purely geometric expansion would suggest.
The same bookkeeping should apply if the internal space is replaced by another noncommutative algebra, so the argument is really a statement about how two star products combine, not about the canonical structure specifically.

Original abstract

An enveloping algebra valued gauge field is constructed, its components are functions of the Lie algebra valued gauge field and can be constructed with the Seiberg-Witten map. This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,265 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

An enveloping algebra valued gauge field is constructed, its components are functions of the Lie algebra valued gauge field and can be constructed with the Seiberg-Witten map. This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	An enveloping algebra valued gauge field is constructed, its components are functions of the Lie algebra valued gauge field and can be constructed with the Seiberg-Witten map.	contribution 0.89	0.68	0.28	asserted 1.00
s1	This allows the formulation of a dynamics for a finite number of gauge field components on non-commutative spaces.	interpretation 0.72	0.65	0.44	asserted 0.82

What kind of paper the classifier reports this is: theoretical result 0.59 · evidence from proof or derivation 0.96.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
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references and citing sentences
relations to other papers

levels run

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S sections
T body sentences
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R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (18)

[6]

B. Jurˇ co, P. Schupp and J. Wess, Nonabelian noncommutative gauge theory , in preperation

View on Pith
[4]

B. Jurˇ co, P. Schupp, Noncommutative Yang-Mills from equivalence of star prod- ucts, Eur. Phys. J. C 14 , 367 (2000), hep-th/0001032

View on Pith Pith review arXiv 2000
[5]

B. Jurˇ co, P. Schupp and J. Wess, Noncommutative gauge theory for Poisson man- ifolds, Nucl. Phys. B, in press, hep-th/0005005

View on Pith Pith review arXiv
[1]

J. Madore, S. Schraml, P. Schupp and J. Wess, Gauge theory on noncommutative spaces, Eur. Phys. J. C, in press, hep-th/0001203

View on Pith Pith review arXiv
[2]

A. Connes, M. R. Douglas, A. Schwarz, Noncommutative Geometry and Matrix Theory: Compactiﬁcation on Tori , JHEP 9802, 003 (1998), hep-th/9711162

View on Pith Pith review arXiv 1998
[3]

N. Seiberg and E. Witten, String theory and noncommutative geometry , JHEP 9909 (1999) 032, hep-th/9908142

View on Pith Pith review arXiv 1999
[7]

H. Weyl, Quantenmechanik und Gruppentheorie , Z. Physik 46, 1 (1927); The theory of groups and quantum mechanics , Dover, New-York (1931), translated from Gruppentheorie und Quantenmechanik , Hirzel Verlag, Leipzig (1928)

View on Pith 1927
[8]

E. P. Wigner, Quantum corrections for thermodynamic equilibrium , Phys. Rev. 40, 749 (1932)

View on Pith 1932
[9]

J. E. Moyal, Quantum mechanics as a statistical theory , Proc. Cambridge Phil. Soc. 45, 99 (1949)

View on Pith 1949
[10]

F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Ste rnheimer, Deformation theory and quantization. I. Deformations of symplectic stru ctures, Ann. Physics 111, 61 (1978)

View on Pith 1978
[11]

M. Kontsevitch, Deformation quantization of Poisson manifolds, I, q-alg/9709040

View on Pith Pith review arXiv
[12]

D. Sternheimer, Deformation Quantization: Twenty Years After , math/9809056

View on Pith Pith review arXiv
[13]

J. Wess and B. Zumino, Covariant diﬀerential calculus on the quatum hyperplane , Nucl. Phys. Proc. Suppl. 18B (1991) 302

View on Pith 1991
[14]

J. Wess, q-deformed Heisenberg Algebras , in H. Gausterer, H. Grosse and L. Pit- tner, eds., Proceedings of the 38. Internationale Universi t¨ atswochen f¨ ur Kern- und Teilchenphysik, no. 543 in Lect. Notes in Phys., Springer-V erlag, 2000, Schlad- ming, January 1999, math-ph/9910013. 12

View on Pith Pith review arXiv 2000
[15]

J. Gomis, T. Mehen and M. B. Wise, Quantum Field Theories With Compact Noncommutative Extra Dimensions , hep-th/0006160

View on Pith Pith review arXiv
[16]

L. Bonora, M. Schnabl, M. M. Sheikh-Jabbari and A. Tomas iello, Noncommutative SO(n) and Sp(n) gauge theories , hep-th/0006091

View on Pith Pith review arXiv
[17]

P.-M. Ho, Twisted Bundle on Noncommutative Space and U(1) Instanton , hep- th/0003012

View on Pith
[18]

L. Alvarez-Gaume, S. R. Wadia, Gauge Theory on a Quantum Phase Space , hep- th/0006219. 13

View on Pith

Forward citations

Cited by 11 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A Gravity Theory on Noncommutative Spaces
hep-th 2005-04 conditional novelty 7.0 of 10

A deformed Hopf algebra of diffeomorphisms on constant-theta noncommutative space supports a covariant tensor calculus and an invariant theta-deformed Einstein-Hilbert action.
Construction of non-Abelian gauge theories on noncommutative spaces
hep-th 2001-04 accept novelty 7.0 of 10

Gauge parameters, potentials, matter fields and field strengths of a non-Abelian gauge theory on a noncommutative space are constructed explicitly to second order in the noncommutativity parameter, yielding calculable...
On a non-geometric approach to noncommutative gauge theories
hep-th 2025-08 conditional novelty 6.0 of 10

A non-geometric bootstrap on the Moyal plane produces noncommutative gauge theories and derives the U(N) fundamental restriction from the requirement that the conserved non-local current be Lie algebra valued.
The Standard Model on Non-Commutative Space-Time
hep-ph 2001-11 accept novelty 6.0 of 10

Expanding the Standard Model on non-commutative space-time to first order in theta yields new quark-gluon-electroweak vertices, parity violation in non-commutative QCD, and an unmodified photon sector in the minimal c...
Noncommutative D=4 gravity coupled to fermions
hep-th 2009-02 conditional novelty 5.0 of 10

A noncommutative extension of first-order gravity coupled to fermions is constructed, with a theta-dependent charge conjugation condition that ensures the commutative limit is ordinary gravity.

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kappa-Minkowski spacetime as the result of Jordanian twist deformation
math-ph 2008-12 conditional novelty 5.0 of 10

Two twist families generate the same kappa-Minkowski noncommutative spacetime but different deformed dispersion relations, with the minimal Jordanian case giving a linear energy-dependent mass shift.
Twisted Gauge Theories
hep-th 2006-03 conditional novelty 5.0 of 10

Twisting the gauge coproduct yields consistent noncommutative gauge theories with enveloping algebra valued connections, whose extra vector fields decouple as the deformation parameter goes to zero.
Quantum Electrodynamics on Noncommutative Spacetime
hep-th 2006-04 reject novelty 4.0 of 10

A proposal to quantize noncommutative U(1) gauge theory before Seiberg-Witten expansion, giving one-loop renormalizability and the ordinary QED value for the spin-dependent g-2.
Deformed Coordinate Spaces; Derivatives
hep-th 2004-08 conditional novelty 3.0 of 10

Derivatives on theta-deformed coordinate spaces, chosen with commuting commutators, generate a deformed Poincare bialgebra under which the Klein-Gordon and Dirac equations remain covariant.
Noncommutative Gauge Theories and Gravity
hep-th 2019-07 unverdicted novelty 2.0 of 10

The paper reviews gauge-theoretic formulations of gravity in ordinary and noncommutative spaces based on the authors' earlier works.
Quantum Field Theory on Noncommutative Spaces
hep-th 2001-09 accept novelty 2.0 of 10

The paper is a self-contained review that organizes noncommutative quantum field theory and its string-theoretic connections into a single pedagogical reference.

How later papers used it

50 later papers cite it

Evidence: some. 49 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 9 of the 71 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

2 later papers by other authors are listed below: 2 used its method or tool.

Uses its method or tool 2 of 9 papers with this label

Used the enveloping algebra to realize the nonabelian group as in this paper and chose a minimal expression of gauge expansion.

Probing $e^{+}e^{-}$ annihilation in noncommutative electroweak model (2008) · 0.93 · paraphrased: the licence does not allow quoting
Followed the Seiberg-Witten method and this paper's enveloping algebra approach to formulate a noncommutative U(1) gauge theory coupled to fermionic matter.

A twisted look on kappa-Minkowski: U(1) gauge theory (2011) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

46 arXiv papers cite this in 65 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 41 sentences without a confident label

Builds on it 2 papers

This idea was further developed in [ref] where it was shown that any commutative (but possibly nonabelian) gauge algebra can be extended to a noncommutative version by a generalization of the Seiberg-Witten map.

Quantum Space-Time and Noncommutative Gauge Field Theories · The Seiberg-Witten map · 0.93
[ref] for non-supersymmetric gauge theories: now the noncommutative vector superfields will be valued in the enveloping algebra of the Lie algebra of the ordinary gauge group.

The Seiberg-Witten map and supersymmetry · Summary and Conclusions · 0.82

Uses its method or tool 9 papers

To construct the NC extension of the standard model (SM) [ref] , which uses the same gauge group and particle content, or for that matter any other gauge theory, be it abelian or non-abelian, one expands the NC gauge fields in non linear power series of $Θ$ [ref] : λ_α(x,Θ) = α(x) + Θ^μν λ^(1)_μν(x;α) + Θ^μν Θ^ησ λ^(2)_μνησ(x;α) + ⋯ A_ρ(x,Θ) = A_ρ(x) + Θ^μν A^(1)_μνρ(x) + Θ^μν Θ^ησ A^(2)_μνησρ(x) + ⋯.

TeV Scale Implications of Non Commutative Space time in Laboratory Frame with Polarized Beams · Non Commutative Standard Model · 0.87
However the Seiberg-Witten map [ref] can be used to express all fields in terms of the classical one, ending up with a finite number of fields.], and is consistent with the $⋆$-gauge transformations only if the gauge parameter satisfies the charge conjugation condition [ref] : C ϵ^_θC = ϵ^T_-θ The NC Majorana condition ([ref]) is consistent also with $⋆$-supersymmetry transformations if the supersymmetry parameter is Majorana, and the bosonic fields satisfy the charge conjugation conditions C ^_θC = ^T_-θ, C V^_θC = V^T_-θ

Noncommutative supergravity in D=3 and D=4 · Local $⋆$-supersymmetry · 0.80
Within the enveloping-algebra framework of Refs. [ref] , one introduces at least a noncommutative field for each ordinary field.

SO(10) GUTs with large tensor representations on Noncommutative Space-time · Introducing the noncommutative fields of the noncommutative SO(10) GUT. · 0.91

Compares with it 1 paper

Note that the action [ref] has the same form as the expanded action for the NC gauge field in the case of $θ$-constant deformation [ref] .

Gauge Theory on Twisted $\kappa$-Minkowski: Old Problems and Possible Solutions · Discussion · 0.99

Cites it as background 10 papers

The SW map [ref] was initially developed for $U(N)$ gauge fields and Moyal-Groenewold $⋆$-products (constant noncommutativity), then applied to gauge fields of arbitrary gauge groups and extended to matter fields in [ref] .

Noncommutative gravity at second order via Seiberg-Witten map · Introduction · 0.91
A dozen years ago, it was put forward a formalism –the enveloping-algebra formalism– which can be used to construct noncommutative gauge theories that are deformations of ordinary theories with simple gauge groups in arbitrary unitary representations [ref] .

Computing the \theta-exact Seiberg-Witten map for arbitrary gauge groups · Introduction · 0.94
In the concept of noncommutive spacetime, there are three kinds of structure [ref] considered: (1) canonical structure, (2) Lie algebra structure, and (3) quantum space structure.

Probing $e^{+}e^{-}$ annihilation in noncommutative electroweak model · Introduction · 0.91

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [6]

B. Jurˇ co, P. Schupp and J. Wess, Nonabelian noncommutative gauge theory , in preperation

View on Pith

[2] [4]

B. Jurˇ co, P. Schupp, Noncommutative Yang-Mills from equivalence of star prod- ucts, Eur. Phys. J. C 14 , 367 (2000), hep-th/0001032

View on Pith Pith review arXiv 2000

[3] [5]

B. Jurˇ co, P. Schupp and J. Wess, Noncommutative gauge theory for Poisson man- ifolds, Nucl. Phys. B, in press, hep-th/0005005

View on Pith Pith review arXiv

[4] [1]

J. Madore, S. Schraml, P. Schupp and J. Wess, Gauge theory on noncommutative spaces, Eur. Phys. J. C, in press, hep-th/0001203

View on Pith Pith review arXiv

[5] [2]

A. Connes, M. R. Douglas, A. Schwarz, Noncommutative Geometry and Matrix Theory: Compactiﬁcation on Tori , JHEP 9802, 003 (1998), hep-th/9711162

View on Pith Pith review arXiv 1998

[6] [3]

N. Seiberg and E. Witten, String theory and noncommutative geometry , JHEP 9909 (1999) 032, hep-th/9908142

View on Pith Pith review arXiv 1999

[7] [7]

H. Weyl, Quantenmechanik und Gruppentheorie , Z. Physik 46, 1 (1927); The theory of groups and quantum mechanics , Dover, New-York (1931), translated from Gruppentheorie und Quantenmechanik , Hirzel Verlag, Leipzig (1928)

View on Pith 1927

[8] [8]

E. P. Wigner, Quantum corrections for thermodynamic equilibrium , Phys. Rev. 40, 749 (1932)

View on Pith 1932

[9] [9]

J. E. Moyal, Quantum mechanics as a statistical theory , Proc. Cambridge Phil. Soc. 45, 99 (1949)

View on Pith 1949

[10] [10]

F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, D. Ste rnheimer, Deformation theory and quantization. I. Deformations of symplectic stru ctures, Ann. Physics 111, 61 (1978)

View on Pith 1978

[11] [11]

M. Kontsevitch, Deformation quantization of Poisson manifolds, I, q-alg/9709040

View on Pith Pith review arXiv

[12] [12]

D. Sternheimer, Deformation Quantization: Twenty Years After , math/9809056

View on Pith Pith review arXiv

[13] [13]

J. Wess and B. Zumino, Covariant diﬀerential calculus on the quatum hyperplane , Nucl. Phys. Proc. Suppl. 18B (1991) 302

View on Pith 1991

[14] [14]

J. Wess, q-deformed Heisenberg Algebras , in H. Gausterer, H. Grosse and L. Pit- tner, eds., Proceedings of the 38. Internationale Universi t¨ atswochen f¨ ur Kern- und Teilchenphysik, no. 543 in Lect. Notes in Phys., Springer-V erlag, 2000, Schlad- ming, January 1999, math-ph/9910013. 12

View on Pith Pith review arXiv 2000

[15] [15]

J. Gomis, T. Mehen and M. B. Wise, Quantum Field Theories With Compact Noncommutative Extra Dimensions , hep-th/0006160

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