Pith. sign in

REVIEW 3 major objections 6 minor 81 references

Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper shows that coreductivity restricts Lorentzian quantum deformations to r-matrices $r\subseteq\beta\,\mathfrak h\wedge\mathfrak t$, admitting $\kappa$-Poincaré duals in any dimension but $\kappa$-(A)dS duals only in (2+1).

desk verdict Useful new definitions (coreductivity/cosymmetry) with a sound algebraic core; the Lorentzian classification needs a qualifier and the closedness of H⊥ needs attention. read the letter →

arxiv 1909.01000 v2 pith:LOJV6ST3 submitted 2019-09-03 math-ph gr-qchep-thmath.MP

Angel Ballesteros , Ivan Gutierrez-Sagredo , Flavio Mercati This is my paper · ORCID
classification math-phgr-qchep-thmath.MP MSC 53D1753C3017B6281R5022E70
keywords PoissonhomogeneousspacesLiebialgebrascoisotropicsubgroupscomplementarydualcoreductivitycosymmetryκ-deformationnoncommutativespacetimes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper develops a duality for Poisson homogeneous spaces: from a quotient $M=G/H$ whose Poisson structure comes from a coisotropic Lie bialgebra on $\mathfrak g$, it constructs a complementary dual space $M^\perp=G^*/H^\perp$ using the dual Lie group and the annihilator of $\mathfrak h$. It introduces two refinements of coisotropy—coreductivity and cosymmetry—which are exactly the conditions under which $M^\perp$ is a reductive or symmetric homogeneous space. For Lorentzian Lie algebras the paper shows that coreductive structures must come from r-matrices of the form $r\subseteq \beta\,\mathfrak h\wedge\mathfrak t$, so the $\kappa$-Poincaré deformation is coreductive in every dimension while the $\kappa$-(A)dS deformation is coreductive only in (2+1) dimensions. This matters because coreductivity is what allows the representation theory of the dual Lie algebra to yield physically consistent uncertainty relations for the noncommutative coordinates of the quantum spacetime.

What carries the argument

The load-bearing object is the cocommutator $\delta$ of the Lie bialgebra, decomposed according to the isotropy subalgebra $\mathfrak h$ and its complement $\mathfrak t$: coisotropy kills $\mathfrak t\wedge\mathfrak t$ in $\delta(\mathfrak h)$, coreductivity kills $\mathfrak h\wedge\mathfrak t$ in $\delta(\mathfrak t)$, and cosymmetry kills $\mathfrak h\wedge\mathfrak h$ in $\delta(\mathfrak h)$. Dualizing swaps $\mathfrak h\leftrightarrow\mathfrak h^\perp$ and $\mathfrak t\leftrightarrow\mathfrak t^\perp$, so these three conditions are exactly the ad-invariance and bracket split conditions that make $M^\perp=G^*/H^\perp$ reductive or symmetric. In the Lorentzian case the paper uses the coboundary form $\delta(X)=[X\otimes 1+1\otimes X,r]$ and shows that the generic decomposition $r\subseteq \alpha\,\mathfrak h\wedge\mathfrak h\oplus\beta\,\mathfrak h\wedge\mathfrak t\oplus\gamma\,\mathfrak t\wedge\mathfrak t$ reduces to $r\subseteq\beta\,\mathfrak h\wedge\mathfrak t$ for coisotropic plus coreductive structures; the modified classical Yang-Baxter equation then constrains the surviving coefficients. The geometry of $M^\perp$—which in general has no $G^*$-invariant metric—is handled through K-structures and the canonical connection of the reductive dual, whose torsion and curvature the paper computes from $\mathfrak g^*$ brackets.

What would settle it

Look for a Lorentzian r-matrix with a nonzero $\mathfrak h\wedge\mathfrak h$ component that satisfies the modified classical Yang-Baxter equation and compute whether $\delta(\mathfrak t)$ contains any $\mathfrak h\wedge\mathfrak t$ term; if such an r-matrix exists with no $\mathfrak h\wedge\mathfrak t$ in $\delta(\mathfrak t)$, the 'coreductive iff $\alpha=0$' theorem fails. Equivalently, a (3+1) Lorentzian Lie bialgebra with $\Lambda\neq0$ that is coreductive with respect to $\mathfrak h$ would falsify the exclusion of the $\kappa$-(A)dS deformation.

Watch

Extended reading notes

Core claim

The central claim is that coisotropy, the condition $\delta(\mathfrak h)\subseteq \mathfrak h\wedge\mathfrak g$ that lets a Poisson-Lie structure on $G$ descend to $M=G/H$, has two natural sharpenings: coreductivity, $\delta(\mathfrak t)\subseteq \mathfrak h\wedge\mathfrak h\oplus \mathfrak t\wedge\mathfrak t$, and cosymmetry, $\delta(\mathfrak h)\subseteq \mathfrak h\wedge\mathfrak t$. These are precisely the conditions that make the complementary dual $M^\perp=G^*/H^\perp$ reductive and symmetric, respectively, and the construction is self-dual: the dual of $M^\perp$ is again $M$. For the Lorentzian Lie algebras $\mathfrak g_\Lambda$—Minkowski and (Anti-)de Sitter in (2+1) and (3+1) dimensions—the paper computes that a coboundary Lie bialgebra with r-matrix $r\subseteq \alpha\,\mathfrak h\wedge\mathfrak h\oplus\beta\,\mathfrak h\wedge\mathfrak t\oplus\gamma\,\mathfrak t\wedge\mathfrak t$ is coisotropic iff $\gamma=0$ and coreductive iff $\alpha=0$, hence coreductive Lorentzian Lie bialgebras are exactly those generated by $r\subseteq \beta\,\mathfrak h\wedge\mathfrak t$. Consequently the $\kappa$-Poincaré Lie bialgebra is coreductive in any dimension, while the $\kappa$-(A)dS Lie bialgebra is coreductive only in (2+1); in the (3+1) Minkowski case the dual space $M^\perp_0$ is six-dimensional with an undeformed Poisson version of $\mathrm{so}(3,1)$, and in (2+1) the dual Poisson brackets are $\Lambda$-deformations of $\mathrm{so}(2,1)$. Coreductivity also guarantees that uncertainty relations between quantum spacetime coordinates take the form $\Delta\hat x\,\Delta\hat\xi\ge \tfrac12\langle\hat\xi\rangle$ rather than involving $\hat x$ terms, which is what makes the representations of the full quantum group restrict cleanly to the spacetime subalgebra.

Load-bearing premise

The construction of the complementary dual as a coset space requires the annihilator subgroup $H^\perp$ to be closed in $G^*$; the paper neither proves nor states this closedness, and if $H^\perp$ fails to be closed then $G^*/H^\perp$ may not be a smooth manifold.

Editorial extensions

If this is right

  • A Lorentzian Poisson homogeneous spacetime admits a reductive complementary dual only when its Lie bialgebra is generated by an r-matrix contained in $\mathfrak h\wedge\mathfrak t$; any deformation with an $\mathfrak h\wedge\mathfrak h$ component is excluded from the dual-reductive framework.
  • The $\kappa$-Poincaré deformation has a coreductive complementary dual in every dimension; in (3+1) dimensions the dual space $M^\perp_0$ is six-dimensional and its Poisson structure is an undeformed Poisson copy of the Lorentz algebra $\mathrm{so}(3,1)$.
  • The $\kappa$-(A)dS deformation admits a reductive complementary dual only in (2+1) dimensions; in (3+1) dimensions the presence of the cosmological constant obstructs coreductivity, so this quantum spacetime drops out of the duality framework.
  • When coreductivity holds, the crossed commutation rules of the dual Lie algebra satisfy $[\hat x,\hat\xi]\subseteq \hat\xi$, so the uncertainty relations take the form $\Delta\hat x\,\Delta\hat\xi\ge \tfrac12\langle\hat\xi\rangle$ and states with $\hat\xi|\psi\rangle=0$ impose no singular constraint on the spacetime-coordinate expectation values.
  • Although the dual spaces generally admit no $G^*$-invariant metric, coreductivity gives them a canonical connection from a K-structure; for the (2+1) $\kappa$-bialgebra the curvature components are proportional to $\Lambda$ (Ricci flat only for $\Lambda=0$), and the (3+1) $\kappa$-Minkowski dual is flat.

Reading between the lines

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

  • The paper's selection rule—only r-matrices of the form $\mathfrak h\wedge\mathfrak t$ survive coreductivity—suggests a general criterion for kinematical Lie algebras beyond Lorentz: if applied to Galilean, Carroll, or Newton-Hooke deformations, it may single out one preferred noncommutative spacetime per kinematical family, which would be testable against existing classifications.
  • The dual space in the (2+1) $\kappa$ case carries a Poisson structure that is a $\Lambda$-deformation of $\mathrm{so}(2,1)$, so one can read $M^\perp$ as a curved momentum or angular-momentum geometry with $\Lambda$ playing the role of curvature; the canonical-connection curvature computed in the paper reinforces that interpretation.
  • Because the duality is self-dual, quantization of the dual Poisson homogeneous space should produce the dual quantum group from the other side; this suggests a testable symmetry: uncertainty relations for coordinates on one side should correspond to uncertainty relations for the Lorentz-sector observables on the dual side, a feature the paper leaves implicit.
  • The closedness of $H^\perp$ is never checked, so a concrete follow-up is to verify for the explicit $\kappa$-Minkowski duals whether the annihilator subgroup is closed in $G^*$; if not, the smooth-quotient status of these physical examples would need separate justification.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper develops a duality framework for Poisson homogeneous spaces (PHS). Starting from a Poisson-Lie group (G,Π) with Lie bialgebra (g,δ) and a coisotropic subgroup H, the authors define the complementary dual space M⊥=G*/H⊥, where H⊥ has Lie algebra the annihilator h⊥⊂g*. They then introduce two new conditions on δ relative to a splitting g=h⊕t: coreductivity, δ(t)⊆h∧h⊕t∧t, which makes M⊥ a reductive homogeneous space for G*; and cosymmetry, δ(h)⊆h∧t, which makes M⊥ symmetric. These conditions are characterized algebraically and applied to Lorentzian Lie algebras gΛ (Minkowski and (A)dS) with h=so(3,1) and t the translation sector. For the κ-deformation, the paper constructs the dual spaces M⊥Λ in (2+1) and (3+1) dimensions, analyzes their Poisson structures and K-structure geometry, and argues that coreductivity controls the uncertainty relations obtained from representations of the dual algebra g*. The central claims are that coreductive Lorentzian Lie bialgebras are exactly those generated by r-matrices r⊆β h∧t, and that the κ-Poincaré bialgebra is coreductive in any dimension while the κ-(A)dS bialgebra is coreductive only in (2+1) dimensions.

Significance. If the main claims hold, the paper provides a clean algebraic criterion for when the complementary dual of a Poisson homogeneous spacetime is reductive or symmetric, with direct consequences for the κ-deformation: the (3+1) κ-(A)dS deformation would be excluded from the duality framework, while κ-Poincaré is included. The algebraic derivation of coreductivity and cosymmetry from the reductive and symmetric conditions in Sections 5.1 and 5.2 is straightforward and checkable, and the explicit computations for the κ-bialgebra are concrete and reproducible. The paper contains no fitted parameters and the framework is self-dual, which is a genuine conceptual strength. However, the Lorentzian classification in Section 6.1 — the load-bearing result — rests on omitted computations, and the definition of M⊥ in Definition 12 presupposes closedness of H⊥ without proof. These gaps currently prevent the paper from fully supporting its strongest claims.

major comments (3)
  1. [§6.1, Eqs. (38)–(41)] The classification of coreductive Lorentzian Lie bialgebras is the paper's main technical result, but it is asserted after 'lengthy but straightforward computations involving explicitly the structure constants ... that we omit here for the sake of brevity'. This is not sufficient for a classification on which the exclusion of the (3+1) κ-(A)dS case and the inclusion of κ-Poincaré depend. The schematic inclusions (39)–(40) are subspace inclusions, not coefficient equations; to conclude that coisotropy forces γ=0 and coreductivity forces α=0 for coisotropic bialgebras, one must verify that the displayed sectors are the only ones, that no cancellation between the α h∧t and γΛ h∧t contributions can occur for generic r, and that the cited classifications in [20,21,58,59] indeed cover all Lorentzian Lie bialgebras, including any non-coboundary ones. Please include the full computations or a complete proof in an appendix.
  2. [Definition 12 (Section 4)] The complementary dual space M⊥=G*/H⊥ is defined by taking 'the unique connected and simply-connected Lie subgroup H⊥ of G* with Lie algebra h⊥'. Two issues arise. First, a connected Lie subgroup with a prescribed Lie algebra need not be closed, and G*/H⊥ is a smooth homogeneous space (as used throughout Sections 6–8) only when H⊥ is closed; no closedness proof or sufficient condition is given. Second, the subgroup need not be simply connected even when G* is simply connected. The definition should either impose closedness explicitly or prove it for the cases considered; otherwise the existence of M⊥ as a manifold is not established.
  3. [Section 8, Eqs. (65)–(68)] The uncertainty-relations argument assumes that the dual Lie algebra g* carries a C*-algebra structure with unitary irreducible representations. This is a substantial assumption: for a finite-dimensional Lie algebra, unitary representations are by unbounded operators, and Robertson-type inequalities such as (66) and (68) require absolute values and domain qualifications. As written, the conclusion that non-coreductivity forces 'singular constraints' on states with ξ|ψ>=0 is heuristic. If this is intended as a physical motivation, it should be labeled as such; if it is intended as a rigorous statement, the hypotheses on the representations and operator domains must be supplied. This does not affect the algebraic classification, but it is central to the claimed physical interpretation.
minor comments (6)
  1. [Section 5, after Eq. (21)] There is a typo: 'Mofeover' should be 'Moreover'.
  2. [Eqs. (66) and (68)] The right-hand sides of the uncertainty inequalities should be absolute values of the expectation values (or the states chosen so the expectations are nonnegative), and the notation should clarify that [x,ξ] is a Lie bracket in g*, not an operator commutator.
  3. [Definition 12] Replace 'connected and simply-connected Lie subgroup' with 'connected Lie subgroup' and add a closedness hypothesis; as written the phrase is mathematically inaccurate.
  4. [Section 6.1] The statement that 'all (A)dS and Poincaré Lie bialgebras are coboundary' is attributed to [58], which concerns the Poincaré case; please give precise references for the (A)dS cases and for the exhaustiveness of the classification list.
  5. [Section 7, Eqs. (63)–(64)] Please specify the identification TeH⊥M⊥ ≃ t⊥ used in the curvature computations and state explicitly that the connection used is the canonical connection of Eq. (61).
  6. [Section 4.1, Eq. (7)] The notation {x,x}Π, {x,ξ}Π, {ξ,ξ}Π is typographically confusing; write e.g. {xi,xj}Π to make the coordinate indices explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the coreductivity classification is a direct application of the paper's own definitions to the displayed r-matrix decomposition.

full rationale

The central results are not predictions fitted to data and do not reduce to their inputs by construction. Coisotropy (Definition 11) is defined as δ(h) ⊆ h∧g, coreductivity (Definition 13) as δ(t) ⊆ h∧h⊕t∧t, and the Section 6.1 classification substitutes the displayed decomposition of δ(h) and δ(t) from a generic r-matrix (equations 39-40) into those conditions; this is a direct algebraic consequence of the definitions rather than a self-referential validation. The Lorentzian examples use the authors' earlier r-matrices and dual Poisson-Lie brackets (e.g., expressions (42), (53), (52), (56), with citations [24,64,67,68]) as inputs that define the κ-deformation being analysed; the coreductivity conclusions are then checked in the present paper (e.g., the √−Λ J1∧J2 term in (53) prevents coreductivity for Λ ≠ 0). Those citations are explicit, checkable formulas, not an unverified uniqueness theorem invoked to force the conclusion. Section 8's uncertainty-relation discussion is an illustrative application of standard inequalities to the stated commutation relations under an explicitly assumed C*-algebra structure, not a result whose conclusion is presupposed. The paper's genuine weaknesses are non-circular: the exhaustive form of the Section 6.1 classification is asserted after omitted 'lengthy but straightforward' structure-constant computations, and Definition 12 assumes without proof that H⊥ is closed in G*; these are completeness and correctness concerns, not circularity. No fitted parameter is renamed as a prediction, and no load-bearing argument reduces to a self-citation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on standard Lie bialgebra/PL group theory plus four domain assumptions: the coboundary nature of Lorentzian Lie bialgebras (from Ref. [58]), the closedness/integration of h⊥ to a subgroup of G*, the dual PL brackets from the authors' prior papers [64,68], and the existence of C*-algebra representations of g*. No free parameters are fitted to data; z and Λ are physical inputs. No new physical entities are postulated.

assumptions (4)
  • domain assumption All (2+1)- and (3+1)-dimensional (A)dS and Poincaré Lie bialgebras are coboundary, so δ(X)=[1⊗X+X⊗1,r] for an r-matrix of the form r⊂α h∧h ⊕ β h∧t ⊕ γ t∧t.
    Invoked in Section 6.1 to derive the schematic cocommutators (39)-(40) and the classification of coreductive Lorentzian Lie bialgebras. Cited to Ref. [58]; if non-coboundary examples exist, the classification is incomplete.
  • domain assumption The dual group G* is connected and simply connected and the annihilator subalgebra h⊥ integrates to a closed Lie subgroup H⊥, so the coset M⊥=G*/H⊥ is a smooth manifold.
    Used in Definition 12 (Section 4) to define the complementary dual space. Closedness is not proved; without it the quotient may not be a manifold.
  • domain assumption The dual Poisson-Lie bracket Π* on G* is as computed in Refs. [64,68] (by the same authors); the projections π* for the κ-deformation are taken from there.
    Sections 6.2-6.3 rely on these prior computations to exhibit the dual PHS brackets (52) and (56). These are inputs, not derived in this paper.
  • domain assumption The dual Lie algebra g* carries a C*-algebra structure admitting unitary irreducible representations for which the formal uncertainty relations from the commutators hold.
    Section 8 assumes this to draw physical conclusions about uncertainty relations. No construction or guarantee is provided for the solvable dual groups considered.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces." pith.science (2026). https://pith.science/paper/LOJV6ST3

@misc{pith2026190901000,
  author       = {Pith},
  title        = {Pith review of: Coisotropic Lie bialgebras and complementary dual Poisson homogeneous spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LOJV6ST3}},
  note         = {Machine review of arXiv:1909.01000}
}
abstract

Quantum homogeneous spaces are noncommutative spaces with quantum group covariance. Their semiclassical counterparts are Poisson homogeneous spaces, which are quotient manifolds of Lie groups $M=G/H$ equipped with an additional Poisson structure $\pi$ which is compatible with a Poisson-Lie structure $\Pi$ on $G$. Since the infinitesimal version of $\Pi$ defines a unique Lie bialgebra structure $\delta$ on the Lie algebra $\frak g=\mbox{Lie}(G)$, we exploit the idea of Lie bialgebra duality in order to study the notion of complementary dual homogeneous space $M^\perp=G^\ast/H^\perp$ of a given homogeneous space $M$ with respect to a coisotropic Lie bialgebra. Then, by considering the natural notions of reductive and symmetric homogeneous spaces, we extend these concepts to $M^\perp$ thus showing that an even richer duality framework between $M$ and $M^\perp$ arises from them. In order to analyse physical implications of these notions, the case of $M$ being a Minkowski or (Anti-) de Sitter Poisson homogeneous spacetime is fully studied, and the corresponding complementary dual reductive and symmetric spaces $M^\perp$ are explicitly constructed in the case of the well-known $\kappa$-deformation, where the cosmological constant $\Lambda$ is introduced as an explicit parameter in order to describe all Lorentzian spaces simultaneously. In particular, the fact that $M^\perp$ is a reductive space is shown to provide a natural condition for the representation theory of the quantum analogue of $M$ that ensures the existence of physically meaningful uncertainty relations between the noncommutative spacetime coordinates. Finally, despite these dual spaces $M^\perp$ are not endowed in general with a $G^\ast$-invariant metric, we show that their geometry can be described by making use of $K$-structures.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

81 extracted references · 60 canonical work pages

  1. [1]

    Maggiore

    M. Maggiore. The algebraic structure of the generalized uncert ainty principle. Phys. Lett. B , 319(1- 3):83–86, 1993. arXiv:hep-th/9309034, doi:10.1016/0370-2693(93)90785-G

    arXiv 1993

  2. [2]

    Doplicher, K

    S. Doplicher, K. Fredenhagen, and J. E. Roberts. Spacetime qu antization induced by classical gravity. Phys. Lett. B , 331(1-2):39–44, 1994. doi:10.1016/0370-2693(94)90940-7

    doi:10.1016/0370-2693(94)90940-7 1994

  3. [3]

    L. J. Garay. Quantum gravity and minimum length. Int. J. Mod. Phys. A , 10(2):145–165, 1995. arXiv:gr-qc/9403008, doi:10.1142/S0217751X95000085

    arXiv 1995

  4. [4]

    R. J. Szabo. Quantum field theory on noncommutative spaces. Phys. Rep. , 378(4):207–299, 2003. arXiv:hep-th/0109162, doi:10.1016/S0370-1573(03)00059-0

    arXiv 2003

  5. [5]

    M. S. Dijkhuizen and T. H. Koornwinder. Quantum homogeneous s paces, duality and quantum 2-spheres. Geom. Dedicata, 52(3):291–315, 1994. doi:10.1007/BF01278478

    work page doi:10.1007/bf01278478 1994

  6. [6]

    Chari and A

    V. Chari and A. Pressley. A guide to Quantum Groups . Cambridge University Press, Cambridge, 1994

    1994

  7. [7]

    S. Majid. Foundations of quantum group theory . Cambridge University Press, Cambridge, 1995. doi:10.1017/CBO9780511613104

    doi:10.1017/cbo9780511613104 1995

  8. [8]

    S. Majid. Hopf algebras for physics at the Planck scale. Class. Quantum Gravity , 5(12):1587–1606,

    work page

Show all 81 references
  1. [9]

    Podle´ s and S

    P. Podle´ s and S. L. Woronowicz. Quantum deformation of Loren tz group. Commun. Math. Phys. , 130(2):381–431, 1990. doi:10.1007/BF02473358

    1990 doi

  2. [10]

    Lukierski, H

    J. Lukierski, H. Ruegg, A. Nowicki, and V. N. Tolstoy. q-deform ation of Poincar´ e algebra. Phys. Lett. B , 264(3-4):331–338, 1991. doi:10.1016/0370-2693(91)90358-W

    1991 doi

  3. [11]

    Majid and H

    S. Majid and H. Ruegg. Bicrossproduct structure of κ-Poincar´ e group and non-commutative geom- etry. Phys. Lett. B , 334(3-4):348–354, 1994. doi:10.1016/0370-2693(94)90699-8

    1994 doi

  4. [12]

    Zakrzewski

    S. Zakrzewski. Quantum Poincar´ e group related to the κ-Poincar´ e algebra.J. Phys. A: Math. Gen. , 27(6):2075–2082, 1994. doi:10.1088/0305-4470/27/6/030

    1994 doi

  5. [13]

    Ballesteros, F

    A. Ballesteros, F. J. Herranz, M. A. del Olmo, and M. Santande r. Quantum (2+1) kine- matical algebras: a global approach. J. Phys. A: Math. Gen. , 27(4):1283–1297, 1994. doi:10.1088/0305-4470/27/4/021

    1994 doi

  6. [14]

    Podle´ s

    P. Podle´ s. Mathematical Physics Spaces of Quantum SU(2) an d SO(3) Groups. Commun. Math. Phys., 170:1–20, 1995. doi:10.1007/BF02099436

    1995 doi

  7. [15]

    Ballesteros, F

    A. Ballesteros, F. J. Herranz, M. A. del Olmo, and M. Santande r. A new ”null-plane” quantum Poincar´ e algebra. Phys. Lett. B , 351(1-3):137–145, 1995. arXiv:q-alg/9502019, doi:10.1016/0370-2693(95)00386-Y

    1995 arXiv

  8. [16]

    N. Ciccoli. Quantum planes and quantum cylinders from Poisson ho mogeneous spaces. J. Phys. A: Math. Gen. , 29(7):1487–1495, 1996. arXiv:q-alg/9504023, doi:10.1088/0305-4470/29/7/018

    1996 arXiv

  9. [17]

    Bonechi, N

    F. Bonechi, N. Ciccoli, R. Giachetti, E. Sorace, and M. Tarlini. Fre e q-Schr¨ odinger equation from homogeneous spaces of the 2-dim euclidean quantum group. Commun. Math. Phys. , 175(1):161–176,

  10. [18]

    Brzezi´ nski

    T. Brzezi´ nski. Quantum homogeneous spaces as quantum qu otient spaces. J. Math. Phys. , 37(5):2388–2399, 1996. arXiv:q-alg/9509015, doi:10.1063/1.531517

    1996 arXiv

  11. [19]

    Ballesteros, N

    A. Ballesteros, N. R. Bruno, and F. J. Herranz. A non-commut ative Minkowskian spacetime from a quantum AdS algebra. Phys. Lett. B , 574(3-4):276–282, 2003. arXiv:hep-th/0306089, doi:10.1016/j.physletb.2003.09.014

    2003 arXiv

  12. [20]

    Borowiec, J

    A. Borowiec, J. Lukierski, and V. N. Tolstoy. Quantum deforma tions of D=4 Euclidean, Lorentz, Kleinian and quaternionic o*(4) symmetries in unified o(4;C) setting. Phys. Lett. B , 754:176–181,

  13. [21]

    Qua ntum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o*(4) symmetries in unified o(4;C) setting

    A. Borowiec, J. Lukierski, and V. N. Tolstoy. Addendum to “Qua ntum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o*(4) symmetries in unified o(4;C) setting” [Phys. Lett. B 754 (2016) 176?181]. Phys. Lett. B , 770:426–430, 2017. arXiv:1704.06852, doi:10.1016...

    2016 arXiv

  14. [22]

    Mercati and M

    F. Mercati and M. Sergola. Physical constraints on quantum d eformations of spacetime symmetries. Nucl. Phys. B , 933:320–339, 2018. arXiv:1802.09483, doi:10.1016/j.nuclphysb.2018.06.014

    2018 arXiv

  15. [23]

    Ballesteros and F

    A. Ballesteros and F. Mercati. Extended noncommutative Minko wski spacetimes and hybrid gauge symmetries. Eur. Phys. J. C , 78:615, 2018. arXiv:1805.07099, doi:10.1140/epjc/s10052-018-6097-1

    2018 arXiv

  16. [24]

    Ballesteros, I

    A. Ballesteros, I. Gutierrez-Sagredo, and F. J. Herranz. Th e κ-(A)dS noncommutative spacetime. Phys. Lett. B , 796:93–101, 2019. arXiv:1905.12358, doi:10.1016/j.physletb.2019.07.038

    2019 arXiv

  17. [25]

    Karolinsky, K

    E. Karolinsky, K. Muzykin, A. Stolin, and V. Tarasov. Dynamical Yang-Baxter equations, quasi- Poisson homogeneous spaces, and quantization. Lett. Math. Phys. , 71(3):179–197, 2005

    2005

  18. [26]

    Drinfeld

    V. Drinfeld. On Poisson homogeneous spaces of Poisson-Lie gro ups. Theor. Math. Phys. , 95(2):524– 525, 1993. doi:10.1007/BF01017137

    1993 doi

  19. [27]

    Zakrzewski

    S. Zakrzewski. Poisson space-time symmetry and correspond ing elementary systems. In H. D. Doebner and V. K. Dobrev, editors, ‘Quantum Symmetries’, Procs. II Int. Wigner Symp. , page 111, Goslar, 1991

    1991

  20. [28]

    Zakrzewski

    S. Zakrzewski. Poisson homogeneous spaces. Proc. XXX Winter Sch. Theor. Phys. , page 10, 1994. arXiv:hep-th/9412101

    1994 arXiv

  21. [29]

    A. G. Reyman. Poisson Structures Related to Quantum Groups . In L. Castellani and J. Wess, editors, Quantum Groups Their Appl. Physics, Proc. Int. Sch. Phys. ”E nrico Fermi”, volume 127, pages 407–443, Amsterdam, 1996. IOS Press. doi:10.3254/978-1-61499-213-4-407

    1996 doi

  22. [30]

    Etingof and D

    P. Etingof and D. Kazhdan. Quantization of Poisson algebraic gr oups and Poisson homogeneous spaces. In A. Connes, K. Gawedki, and J. Zinn-Justin, editors, Symm´ etries quantiques, (Les Houches 1995), page 935, Amsterdam, 1998. North-Holland. arXiv:q-alg/9510020

    1995 arXiv

  23. [31]

    Ciccoli and F

    N. Ciccoli and F. Gavarini. A quantum duality principle for coisotro pic subgroups and Poisson quotients. Adv. Math. (N. Y). , 199(1):104–135, 2006. arXiv:math/0603376, doi:10.1016/j.aim.2005.01.009

    2006 arXiv

  24. [32]

    Ballesteros, C

    A. Ballesteros, C. Meusburger, and P. Naranjo. AdS Poisson h omogeneous spaces and Drinfel’d doubles. J. Phys. A: Math. Theor. , 50(39):395202, 2017. arXiv:1701.04902v3, doi:10.1088/1751-8121/aa858c

    2017 arXiv

  25. [33]

    Ballesteros, I

    A. Ballesteros, I. Gutierrez-Sagredo, and F. J. Herranz. No ncommutative spaces of worldlines. Phys. Lett. B , 792:175–181, 2019. arXiv:1902.09132, doi:10.1016/j.physletb.2019.03.029

    2019 arXiv

  26. [34]

    Ballesteros, G

    A. Ballesteros, G. Gubitosi, I. Gutierrez-Sagredo, and F. J. H erranz. The κ-Newtonian and κ-Carrollian algebras and their noncommutative spacetimes. Phys. Lett. B , 805:135461, 2020. arXiv:2003.03921, doi:10.1016/j.physletb.2020.135461

    2020 arXiv

  27. [35]

    Karolinsky

    E. Karolinsky. Symplectic leaves on Poisson homogeneous space s of Poisson-Lie groups. Mat. Fiz. Anal. Geom., 2(3-4):306–311, 1995

    1995

  28. [36]

    Karolinsky

    E. Karolinsky. Classification of Poisson homogeneous spaces of a compact Poisson-Lie group. Mat. Fiz. Anal. Geom. , 3(3-4):274–289, 1996

    1996

  29. [37]

    Karolinsky

    E. Karolinsky. On the classification of Lagrangian subalgebras in g×g, where g is a complex reductive Lie algebra. Mat. Fiz. Anal. Geom. , 4(1-2):65–74, 1997

    1997

  30. [38]

    Karolinsky

    E. Karolinsky. A classification of Poisson homogeneous spaces o f complex reductive Poisson-Lie groups. In Poisson Geom. , pages 103–108. Warsaw, 1998

    1998

  31. [39]

    J. Lu. Classical dynamical r-matrices and homogeneous Poisso n structures on G/H and K/T. Com- mun. Math. Phys. , 212(2):337–370, 2000. arXiv:math/9909004, doi:10.1007/s002200000209

    arXiv

  32. [40]

    J. Lu. Multiplicative and Affine Poisson Structures on Lie Groups . PhD thesis, 1990. 25

    1990

  33. [41]

    S. S. Chern. Pseudo-groupes continus infinis. In G´ eom´ etrie diff´ erentielle, volume 52, pages 119–136, Strasbourg, 1953

    1953

  34. [42]

    Kobayashi and K

    S. Kobayashi and K. Nomizu. Foundations of differential geometry. Volume II . Interscience Pub- lishers, 1969

    1969

  35. [43]

    Lukierski, Z

    J. Lukierski, Z. ˇSkoda, and M. Woronowicz. κ-deformed covariant quantum phase spaces as Hopf algebroids. Phys. Lett. B , 750:401–406, 2015. arXiv:1507.02612, doi:10.1016/j.physletb.2015.09.042

    2015 arXiv

  36. [44]

    Gutierrez-Sagredo

    I. Gutierrez-Sagredo. Lorentzian Poisson homogeneous spaces, quantum groups and noncommutative spacetimes. PhD thesis, University of Burgos, 2019

    2019

  37. [45]

    Borowiec and A

    A. Borowiec and A. Pachol. κ-Minkowski spacetime as the result of Jordanian twist deformation . Phys. Rev. D , 79(4):04501, 2009. arXiv:0812.0576, doi:10.1103/PhysRevD.79.045012

    2009 arXiv

  38. [46]

    Aschieri, A

    P. Aschieri, A. Borowiec, and A. Pachol. Observables and disper sion relations in κ- Minkowski spacetime. J. High Energy Phys. , 2017(10):152, 2017. arXiv:1703.08726, doi:10.1007/JHEP10(2017)152

    2017 arXiv

  39. [47]

    Borowiec and A

    A. Borowiec and A. Pachol. κ-Deformations and Extended κ-Minkowski Spacetimes. Symmetry, Integr. Geom. Methods Appl. , 10, nov 2014. doi:10.3842/SIGMA.2014.107

    2014 doi

  40. [48]

    Drinfeld

    V. Drinfeld. Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations. Sov. Math. Dokl. , 27:68–71, 1983

    1983

  41. [49]

    Drinfeld

    V. Drinfeld. Quantum Groups. Int. Congr. Math. , pages 798–820, 1987. http://www.ulb.tu-darmstadt.de/tocs/31407404.pdf

    1987

  42. [50]

    M. A. Semenov-Tian-Shanskii. Poisson-Lie groups. The quantu m duality principle and the twisted quantum double. Theor. Math. Phys. , 93(2):1292–1307, 1992. arXiv:hep-th/9304042, doi:10.1007/BF01083527

    1992 arXiv

  43. [51]

    Gavarini

    F. Gavarini. The quantum duality principle. Ann. l’institut Fourier , 52(3):809–834, 2002. arXiv:math/9909071, doi:10.5802/aif.1902

    2002 arXiv

  44. [52]

    Karolinsky

    E. Karolinsky. Classification of Poisson homogeneous spaces of compact Poisson-Lie groups. Dokl. Akad. Nauk , 359(1):13–15, 1998

    1998

  45. [53]

    Karolinsky and A

    E. Karolinsky and A. Stolin. Classical dynamical r-matrices, Pois son homogeneous spaces, and Lagrangian subalgebras. Lett. Math. Phys. , 60(3):257–274, 2002. arXiv:math/0110319v2, doi:10.1023/A:1016231526203

    2002 arXiv

  46. [54]

    Karolinsky and K

    E. Karolinsky and K. Muzykin. On quasi-Poisson homogeneous sp aces of quasi-Poisson Lie groups. J. Lie Theory , 14(2):543–554, 2004

    2004

  47. [55]

    Zakrzewski

    S. Zakrzewski. Poisson homogeneous spaces. In J. Lukierski, Z. Popowicz, and J. Sobczyk, editors, Quantum groups, formalism Appl. Proc. XXX Winter Sch. Theor . Phys. 14?26February 1994 , pages 629–639, Karpacz, 1995. Polish Scientific Publishers PWN

    1994

  48. [56]

    Meusburger and T

    C. Meusburger and T. Sch¨ onfeld. Gauge fixing in (2+1)-gravit y: Dirac bracket and spacetime geometry. Class. Quantum Gravity , 28(12), 2011. arXiv:1012.1835v1, doi:10.1088/0264-9381/28/12/125008

    2011 arXiv

  49. [57]

    Meusburger and T

    C. Meusburger and T. Sch¨ onfeld. Gauge Fixing and Classical Dy namical r-Matrices in ISO(2, 1)-Chern-Simons Theory. Commun. Math. Phys. , 327(2):443–479, 2014. arXiv:1203.5609v2, doi:10.1007/s00220-014-1938-8

    2014 arXiv

  50. [58]

    Zakrzewski

    S. Zakrzewski. Poisson Structures on the Poincar´ e Group. Commun. Math. Phys. , 185(2):285–311,

  51. [59]

    Stachura

    P. Stachura. Poisson-Lie structures on Poincar´ e and Euclide an groups in three dimensions. J. Phys. A: Math. Gen. , 31(19):4555–4564, 1998. doi:10.1088/0305-4470/31/19/018

    1998 doi

  52. [60]

    Ballesteros, F

    A. Ballesteros, F. J. Herranz, and F. Musso. On quantum defo rmations of (anti-)de Sitter al- gebras in (2+1) dimensions. J. Phys. Conf. Ser. , 532(1):012002, 2014. arXiv:1302.0684v2, doi:10.1088/1742-6596/532/1/012002. 26

    2014 arXiv

  53. [61]

    Lukierski, A

    J. Lukierski, A. Nowicki, and H. Ruegg. Real forms of complex qu antum Anti de sitter algebra Uq (Sp(4;C) and their contraction schemes. Phys. Lett. B , 271(3-4):321–328, 1991. arXiv:hep-th/9108018, doi:10.1016/0370-2693(91)90094-7

    1991 arXiv

  54. [62]

    Amelino-Camelia, L

    G. Amelino-Camelia, L. Smolin, and A. Starodubtsev. Quantum sy mmetry, the cosmological constant and Planck-scale phenomenology. Class. Quantum Gravity , 21(13):3095–3110, 2004. arXiv:hep-th/0306134, doi:10.1088/0264-9381/21/13/002

    2004 arXiv

  55. [63]

    Ballesteros, N

    A. Ballesteros, N. R. Bruno, and F. J. Herranz. Non-commuta tive relativistic spacetimes and world- lines from 2+ 1 quantum (anti) de Sitter groups. Adv. High Energy Phys. , 2017:7876942 (19 pages),

    2017

  56. [64]

    Ballesteros, G

    A. Ballesteros, G. Gubitosi, I. Gutierrez-Sagredo, and F. J. H erranz. Curved momentum spaces from quantum groups with cosmological constant. Phys. Lett. B , 773:47–53, 2017. arXiv:1707.09600, doi:10.1016/j.physletb.2017.08.008

    2017 arXiv

  57. [65]

    Maslanka

    P. Maslanka. The n-dimensional κ-Poincar´ e algebra and group. J. Phys. A: Math. Gen. , 26(24):L1251–L1253, 1993. doi:10.1088/0305-4470/26/24/001

    1993 doi

  58. [66]

    Ballesteros and F

    A. Ballesteros and F. Musso. Quantum algebras as quantization s of dual Poisson-Lie groups. J. Phys. A: Math. Theor. , 46(19):195203, 2013. arXiv:1212.3809, doi:10.1088/1751-8113/46/19/195203

    2013 arXiv

  59. [67]

    Ballesteros, N

    A. Ballesteros, N. A. Gromov, F. J. Herranz, M. A. del Olmo, an d M. Santander. Lie bialgebra contractions and quantum deformations of quasi-orthogonal alg ebras. J. Math. Phys. , 36(10):5916– 5937, 1995. arXiv:hep-th/9412083v3, doi:10.1063/1.531368

    1995 arXiv

  60. [68]

    Ballesteros, F

    A. Ballesteros, F. J. Herranz, F. Musso, and P. Naranjo. The κ-(A)dS quantum al- gebra in (3 + 1) dimensions. Phys. Lett. B , 766:205–211, 2017. arXiv:1612.03169, doi:10.1016/j.physletb.2017.01.020

    2017 arXiv

  61. [69]

    Ballesteros, G

    A. Ballesteros, G. Gubitosi, I. Gutierrez-Sagredo, and F. J. H erranz. Curved momentum spaces from quantum (anti-)de Sitter groups in (3+1) dimensions. Phys. Rev. D , 97(10):106024, 2018. arXiv:1711.05050, doi:10.1103/PhysRevD.97.106024

    2018 arXiv

  62. [70]

    W.-K. Tung. Group Theory in Physics . World Scientific, Singapore, 1985. doi:10.1142/0097

    1985 doi

  63. [71]

    Kobayashi and K

    S. Kobayashi and K. Nomizu. Foundations of differential geometry. Volume I . Interscience Publish- ers, 1963

    1963

  64. [72]

    H. Snyder. Quantized space-time. Phys. Rev. , 71(1):38–41, 1947. doi:10.1103/PhysRev.71.38

    1947 doi

  65. [73]

    Doplicher, K

    S. Doplicher, K. Fredenhagen, and J. E. Roberts. The quantu m structure of spacetime at the Planck scale and quantum fields. Commun. Math. Phys. , 172(1):187–220, 1995. arXiv:hep-th/0303037, doi:10.1007/BF02104515

    1995 arXiv

  66. [74]

    Chaichian, P

    M. Chaichian, P. P. Kulish, K. Nishijima, and A. Tureanu. On a Lore ntz-invariant interpretation of noncommutative space-time and its implications on noncommutativ e QFT. Phys. Lett. B , 604(1- 2):98–102, 2004. arXiv:hep-th/0408069, doi:10.1016/j.physletb.2004.10.045

    2004 arXiv

  67. [75]

    Lizzi, M

    F. Lizzi, M. Manfredonia, F. Mercati, and T. Poulain. Localization and reference frames in κ-Minkowski spacetime. Phys. Rev. D , 99(8):085003, 2019. arXiv:1811.08409, doi:10.1103/PhysRevD.99.085003

    2019 arXiv

  68. [76]

    Ballesteros, F

    A. Ballesteros, F. J. Herranz, M. A. del Olmo, and M. Santande r. Classical deformations, Poisson- Lie contractions, and quantization of dual Lie bialgebras. J. Math. Phys. , 36(2):631–640, 1995. doi:10.1063/1.531331. 27

    1995 doi

  69. [1988]

    doi:10.1088/0264-9381/5/12/010

    doi

  70. [1996]

    doi:10.1007/BF02101628

    doi

  71. [1997]

    arXiv:q-alg/9602001, doi:10.1007/s002200050091

    arXiv

  72. [2016]

    arXiv:1511.03653, doi:10.1016/j.physletb.2016.01.016. 24

    2016 arXiv

  73. [2017]

    arXiv:hep-th/0401244, doi:10.1155/2017/7876942

    2017 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.