Pith. sign in

REVIEW 2 major objections 5 minor 98 references

The BMS3 asymptotic symmetry algebra carries a bi-Hamiltonian integrable hierarchy of commuting flows, recovered also as the flat limit of AdS3 and as Lax flows of energy-dependent Schrödinger operators.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-31 06:52 UTC pith:2UXCL5J4

load-bearing objection Solid structural re-proof of the BMS3 hierarchy with a real gap in the flat-case Lenard induction; still worth referee time. the 2 major comments →

arxiv 2607.28454 v1 pith:2UXCL5J4 submitted 2026-07-30 math-ph gr-qcmath.MP

Integrability in Asymptotic Symmetries of Spacetime: the BMS₃ scenario

Corentin Vitel This is my paper
classification math-ph gr-qcmath.MP MSC 37K1037K3017B6883C80
keywords BMS3integrable hierarchybi-Hamiltonian structureNijenhuis operatorasymptotic symmetriesenergy-dependent Schrödinger operatorsAdS3 flat limitLie-Poisson bracket
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper rebuilds and strengthens the claim that the three-dimensional BMS algebra of asymptotic symmetries of flat spacetime supports a genuine infinite-dimensional integrable system. From the algebra itself one constructs a compatible pair of Hamiltonian operators on the space of densities; the resulting recursion produces infinitely many commuting evolution equations whose conserved quantities are in involution. The same construction works for the AdS3 algebra (two copies of KdV) and reduces to the flat case in the flat limit. A parallel Lie-Poisson freezing argument shows the hierarchy is not unique, while the coadjoint orbits of BMS3 are shown to generate the Lax flows of a natural family of energy-dependent Schrödinger operators. The point for a reader is that asymptotic gravitational symmetries are not merely kinematic: they organise an integrable hierarchy whose PDEs can coincide with Einstein dynamics under suitable boundary conditions.

Core claim

The centrally extended bms3 algebra determines a bi-Hamiltonian pair (E, D) on the variational complex; the associated Nijenhuis operator R = E D^{-1} generates an infinite hierarchy of commuting flows K_n = R^n K_0 whose Hamiltonians are in involution for both brackets, so the hierarchy is integrable. The same holds for ads3, and the flat limit recovers the bms3 pair. The Lax flows of 2-energy-dependent Schrödinger operators are precisely the coadjoint Hamiltonian equations on the regular dual of bms3.

What carries the argument

The bi-Hamiltonian pair of Proposition 2.4 (E built from the bms3 coadjoint action, D from the central cocycle) together with the Nijenhuis recursion operator R = E D^{-1}; the Lenard scheme K_n = E dH_{n-1} = D dH_n then produces the integrable hierarchy.

Load-bearing premise

The second operator D must be formally non-degenerate so the recursion operator is well-defined and every higher flow stays in the image of D; if that fails on the actual function space, the infinite commuting hierarchy is not secured.

What would settle it

Compute the first several Lenard recursions explicitly for the flat bms3 pair and check whether the resulting vector fields commute and whether the associated 1-forms are closed; or verify that the flat-limit map from the ads3 pair lands exactly on the bms3 operators of Proposition 2.4.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Under the boundary conditions of the earlier literature, sectors of the 3d Einstein equations become members of this integrable hierarchy and can be solved by integrable-system methods.
  • The freezing-point construction yields a conjectural list of further bms3-like bi-Hamiltonian hierarchies, each potentially corresponding to a different coadjoint orbit type.
  • The ads3 hierarchy is two commuting copies of KdV; its flat limit supplies a controlled deformation from AdS3 integrability to flat-space integrability.
  • Lax pairs for 2-energy-dependent Schrödinger operators are realised geometrically as coadjoint motion on bms3, linking spectral problems to asymptotic symmetry orbits.
  • A tau-scheme generator exists for the flat case, giving an independent inductive construction of the same commuting flows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the fifteen freeze-point classes are integrable, each class may label a distinct integrable sector of 3d gravity with its own boundary charges.
  • The master-symmetry Virasoro algebra generated by the Liouville field suggests a hidden bispectral or Darboux structure for bms3 potentials, analogous to known KdV master symmetries.
  • The quotient description L(vir)+ / r λ² L(vir)+ ≃ bms3 indicates that a full r-matrix construction should produce the hierarchy systematically and may extend to BMS4.
  • Flat-space holography could inherit integrable structure from gca2 ≃ bms3, offering conserved quantities on the celestial circle.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper revisits the bms3-integrable hierarchy of Fuentealba et al. (2018) by constructing a bi-Hamiltonian pair (E,D) on the variational complex from the centrally extended bms3 Lie algebra and Gelfand–Fuks-type cocycles (Prop. 2.4), attaching a Nijenhuis operator R=ED^{-1}, and claiming an integrable Lenard hierarchy (Thm. 2.9) with commuting flows Kn=R^n K0 and Hamiltonians in involution. Parallel constructions are given for ads3 (two KdV copies) with a flat-limit recovery of the bms3 operators, a τ-scheme for the asymptotically flat case (Sec. 2.5), a Lie–Poisson/frozen-point description suggesting non-uniqueness (Conjecture 4.3), and an identification of 2-energy-dependent Schrödinger Lax flows with coadjoint equations on bms3* (Thm. 5.3).

Significance. If the integrability claim holds for the physically relevant (asymptotically flat) hierarchy, the work supplies a systematic structural foundation—variational complex, Nijenhuis recursion, τ-scheme, and Lie–Poisson freezing—linking BMS3 asymptotic symmetries to classical integrable systems, and it opens a concrete route from coadjoint orbits to energy-dependent spectral problems and possibly to other freeze-point hierarchies relevant to 3d Einstein dynamics. The multi-method approach and the explicit flat-limit check from ads3 are genuine strengths; the Lax–coadjoint match (Thm. 5.3) is a clear, checkable contribution independent of the Lenard gap.

major comments (2)
  1. [Appendix; Prop. 2.8; Thm. 2.9; Eqs. 41, 90–91] Theorem 2.9 rests on Prop. 2.8 (Kn=R^n K0 lies in im(D)) so that the Lenard scheme closes and the 1-forms are exact. The Appendix induction uses the splitting R=[[r2,r1],[0,r2]] with r2=J2∂^{-1} and r1=J1∂^{-1}-J2∂^{-1} (Eqs. 90–91), which is exactly ED^{-1} only when D=[[∂,∂],[∂,0]] (both indicators on, c1≠0). In the asymptotically flat case c1=0 one has D=[[0,∂],[∂,0]], so R=[[J2∂^{-1},J1∂^{-1}],[0,J2∂^{-1}]] and the preimage formula (91) (with the combination u1-2u2) does not apply. The written proof therefore does not secure Thm. 2.9 for the main physical hierarchy of Eq. 41. Sec. 2.5 supplies a separate τ-scheme for the flat case, but that does not close the Lenard/Nijenhuis route claimed for Thm. 2.9. The Appendix (or an analogous induction) must be rewritten for c1=0, or the theorem statement must be restricted and the flat case routed entirely through the τ-scheme with a complete
  2. [Sec. 2.3–2.4; Prop. 2.5–2.8; Thm. 2.10] Formal non-degeneracy of D (needed for R=ED^{-1} and for Thm. 2.10) is used throughout. On the circle, ∂ has a kernel of constants and ∂^{-1} is multi-valued; the paper works in the formal symbol calculus but does not state the function space or the quotient by constants/zero modes. For the flat D=[[0,∂],[∂,0]] this is especially delicate. A short precise statement of the formal setting (or of the periodic/mean-zero conventions under which D is invertible on the relevant complement) is needed so that “Nijenhuis” and “image of D” are unambiguous.
minor comments (5)
  1. [Sec. 2.2, Prop. 2.4] Notation for central charges and indicators (1_{c_i≠0}) is introduced late and used inconsistently between the general pair (41) and the flat specialisation; a single convention table early in Sec. 2 would help.
  2. [Sec. 3, Eqs. 63–65] In Sec. 3 the flat-limit operators (65) recover (41) only after identifying c1=lim(c-c̄) and c2=lim l^{-1}(c+c̄) and discarding 1/l^2 terms; this is standard but should be written as an explicit operator limit rather than left implicit.
  3. [Sec. 4, Conjecture 4.3; Conclusions] Conjecture 4.3 lists fifteen freeze-point classes without a derivation or reference for the count; since it is not used for Thm. 2.9 it can stay, but it should be clearly labelled as open and not mixed with proved statements in the conclusions.
  4. [Throughout] Typos and style: “Schr¨ odinger” spacing, “abms3” missing spaces in the abstract/intro, “eH” vs “H̃” inconsistency, and “τ−scheme” hyphenation vary across the text.
  5. [Figure 1; Sec. 5] Figure 1 is helpful but several arrows (e.g. “Opening?” and the r-matrix quotient) are not developed in the body; either expand briefly or mark them as outlook.

Circularity Check

0 steps flagged

No significant circularity: bi-Hamiltonian pair and integrability are read off the known bms3/ads3 Lie data via standard Gel'fand–Dorfman/Magri machinery, not forced by fit or self-citation.

full rationale

The central claims (Prop. 2.4, Thm. 2.9, Thm. 3.3, Prop. 4.1–4.2, Thm. 5.3) are algebraic constructions: the operators E and D are the Kirillov–Kostant/Lie–Poisson structures and Gelfand–Fuks-type cocycles of centrally extended bms3 (resp. two Virasoro copies for ads3); integrability follows from a Nijenhuis/Lenard recursion (and a separate τ-scheme) once D is formally non-degenerate. That is definitional of the Magri scheme applied to a given Lie algebra, not a loop in which a target PDE or Einstein reduction is smuggled back as input. Fuentealba et al. supply historical motivation and the physical reading that Einstein equations match the flows under boundary conditions; they are not used as a uniqueness theorem or as the definition of the bi-Hamiltonian pair proved here. The energy-dependent Schrödinger identification (Thm. 5.3) matches known Antonowicz–Fordy operators to coadjoint bms3 flows rather than renaming an empirical fit. Conjecture 4.3 is explicitly open and not load-bearing. No fitted parameters, no self-citation chain, and no X-defined-as-Y prediction appear. (A separate correctness concern—that the Appendix induction for Prop. 2.8 is written for the c1≠0 form of R—does not constitute circularity under this rubric.)

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 2 invented entities

The paper works in the standard formal calculus of variations and Lie-Poisson geometry on centrally extended bms3/ads3. No experimental fits. Load-bearing inputs are the known Lie brackets/cocycles, Gel'fand-Dorfman theorems equating Hamiltonian operators with Lie structures on 1-forms, non-degeneracy of D for R, and the Antonowicz-Fordy setup for energy-dependent Schrödinger operators. Conjecture 4.3 postulates a classification of freeze points without proof.

free parameters (3)
  • Central charges c1, c2 (and Brown-Henneaux c, c-bar; flat c2 ~ 3/G)
    Taken as fixed structure constants of the extended algebras / physical inputs from 3d gravity; not fitted in this paper. Indicator 1_{c_i≠0} selects degenerate vs non-degenerate D.
  • Seed constants a,b in dH_{-1} = b du1 + a du2 = b in {0,1} for τ-scheme; a,b real otherwise
    Free choice of Casimir seed for the Lenard/τ hierarchy; b restricted to {0,1} in the τ-scheme. Labels families of flows, not data fits.
  • Freeze-point / H1-energy parameters (θ_i, ι_i) and m0 = m0 = (1/2 dx^2, 0; 1/2 dx^2, 0) for main hierarchy
    Choice of frozen coadjoint point that defines the second bracket; natural choice recovers D, other choices conjecturally give other hierarchies.
axioms (6)
  • standard math Gel'fand-Dorfman theorem: a differential matrix operator E is Hamiltonian iff eΩ^1 carries the Lie bracket defined by the structure coefficients c^α_ijkl (Thm 2.3 / GD81).
    Used in §2.2 to read E off the bms3 bracket on 1-forms.
  • standard math Compatibility of E and constant D is equivalent to the bilinear form from D being a 2-cocycle (Dorfman 1993).
    Used to attach the second operator from the central extension cocycle (38)-(41).
  • domain assumption Centrally extended bms3 and ads3 brackets and coadjoint actions as in Def 1.1, Prop 1.2, Def 3.1 (from Barnich-Oblak, Brown-Henneaux, etc.).
    Algebraic starting point for all Hamiltonian structures.
  • domain assumption Formal non-degeneracy of D so R = E D^{-1} exists and Lenard recursion stays in im(D); variational complex exactness in positive degrees (D93) for τ-scheme existence.
    Invoked for Nijenhuis operator, Prop 2.8, Thm 2.9, Thm 2.10.
  • domain assumption Antonowicz-Fordy Lax construction for N=2 energy-dependent Schrödinger operators yields Hamiltonian matrix operators and recursion constraints (AF87-89).
    Section 5 identifies those operators with E and coadjoint bms3 flows.
  • ad hoc to paper Conjecture 4.3: Poisson pairs of codimension 4 on bms3* fall into fifteen freeze-point classes, suggesting non-unique bms3-like hierarchies.
    Stated as conjecture expanding KM02 Virasoro case; not proved; motivates future classification.
invented entities (2)
  • bms3 master symmetries Y_n = R^{n+1} X spanning an upper Virasoro algebra with vanishing central charge no independent evidence
    purpose: Link the pencil's conformal symmetry to master symmetries and possible bispectral/Schrödinger connections.
    Defined from Liouville field X and Nijenhuis R (Props 2.5-2.7); investigation deferred to future work.
  • Fifteen-class freeze-point taxonomy of codimension-4 bms3* Poisson pairs (Conjecture 4.3) no independent evidence
    purpose: Argue the constructed hierarchy is not unique and classify other H1-energy hierarchies.
    Listed constant points in (77); no orbit-by-orbit proof or explicit extra hierarchies constructed.

pith-pipeline@v1.2.0-daily-grok45 · 30690 in / 4693 out tokens · 89769 ms · 2026-07-31T06:52:09.336671+00:00 · methodology

0 comments
read the original abstract

We revisit the proof of a $\mathfrak{bms}_3-$integrable hierarchy introduced in Fuentealba et al. (JHEP, 2018), using different structural methodologies. Specifically, from the variational complex on the ring of polynomial symbols, a $\mathfrak{bms}_3$ bi-Hamiltonian structure is constructed on which a Nijenhuis operator can be attached. A similar argument is made for the $\mathrm{AdS}_3$ case, where we check that the flat limit of the latter recovers the asymptotically flat situation. An alternative $\tau-$scheme description is also presented. Moreover, a Lie-Poisson description suggests that such a $\mathfrak{bms}_3-$hierarchy is not unique. For a subclass of so-called energy-dependent Schr\"odinger operators, it is shown that their Lax flows are described by the coadjoint orbits of $\mathfrak{bms}_3$.

Figures

Figures reproduced from arXiv: 2607.28454 by Corentin Vitel.

Figure 1
Figure 1. Figure 1: Overview of the structures involved in the integrable bms3 hierarchy [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

98 extracted references · 3 linked inside Pith

  1. [1]

    148 (2018)

    Fuentealba, O., Matulich, J., Pérez, A., et al., Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes, J.High Energ.Phys. 148 (2018)

    2018

  2. [2]

    Gardner, C.S., Green, J.M., Kruskal, M.D., Miura, R.M., Method for solving the KdV equation, Phys.Rev.Lett, 19, 1095--1097 (1967)

    1967

  3. [3]

    Lax, P.D., Integrals of nonlinear equations and solitary waves, Commum.Pure Appl.Math., 21, 467--490 (1968)

    1968

  4. [4]

    5, 18--27 (1971)

    Zakharov V.E., Fadeev, L.D., The Korteweg-de Vries equation is a completely integrable Hamiltonian system, Funk.Anal.Priloz. 5, 18--27 (1971)

    1971

  5. [5]

    Gardner, C.S., Korteweg-de Vries Equation and Generalizations IV, J. Math. Phys. 12, 1548–1551 (1971)

    1971

  6. [6]

    19, 1156--1162 (1978)

    Magri, F., A simple model of the integrable Hamiltonian equation, J.Math.Phys. 19, 1156--1162 (1978)

    1978

  7. [7]

    Miwa, T., Jimbo, M., Date, T., Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras, Cambdrige University Press (2000)

    2000

  8. [8]

    Semenov-Tian-Shansky, M., Integrable Sytems and Factorization Problems, arxiv: nlin/0209057, 1--69 (2002)

    Pith/arXiv arXiv 2002

  9. [9]

    Gel'fand, I.M., Dickey, L.A., Lie algebra structure in the formal variational calculus, Funk.Anal.Priloz., 10, 18--25 (1976)

    1976

  10. [10]

    Gel'fand, I.M., Dickey, L.A., Fractional powers of operators and Hamiltonian systems, Funk.Anal.Priloz., 10, 13--29 (1976)

    1976

  11. [11]

    Gel'fand, I.M., Manin, Y.I., Shubin, M.A., Poisson brackets and the kernel of the variational derivative in the formal calculus of variations, Funct.Anal.Its.Appl 10, 274--278 (1976)

    1976

  12. [12]

    Segal, G., Unitary representations of some infinite dimensional groups, Comm.Math.Phys., 80, 301--342 (1981)

    1981

  13. [13]

    16, 2859--2869 (1991)

    Segal, G., The geometry of the KdV equation, International Journal of Modern Physics A, Vol 6, no. 16, 2859--2869 (1991)

    1991

  14. [14]

    Witten, E., Two-Dimensional Gravity and Intersection Theory on Moduli Space, Surveys in Differential Geometry, 1, 243--310 (1991)

    1991

  15. [15]

    Kontsevich, M., Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function, Commun.Math.Phys, 147, 1--23 (1992)

    1992

  16. [16]

    VII Waves from axe-symmetric isolated systems, Proc.Royal Soc., A, Maths.Phys.Eng.Sci., 269 (1962)

    Bondi, H., van der Burg, M.G.J., Metzner, A.W.K., Gravitational waves in general relativity. VII Waves from axe-symmetric isolated systems, Proc.Royal Soc., A, Maths.Phys.Eng.Sci., 269 (1962)

    1962

  17. [17]

    VIII Waves in asymptotically flat space-time, Proc.Royal Soc., A, Math.Phys.Eng.Sci, 270 (1962)

    Sachs, R.K., Gravitational waves in general relativity. VIII Waves in asymptotically flat space-time, Proc.Royal Soc., A, Math.Phys.Eng.Sci, 270 (1962)

    1962

  18. [18]

    Sachs, R.K., Asymptotic symmetries in gravitational theory, Phys.Rev.Lett, 128 (1962)

    1962

  19. [19]

    High Energy.Phys., 5, 1--62 (2010)

    Barnich, G., Troessaert, C., Aspects of the BMS/CFT correspondence, J. High Energy.Phys., 5, 1--62 (2010)

    2010

  20. [20]

    Barnich, G., Troessaert, C., Symmetries of asymptotically flat four-dimensional spacetimes at null infinity revisited, Phy.Rev.Lett, 105 (2010)

    2010

  21. [21]

    Barnich, G., Troessaert, C., BMS charge algebra, J.High Energy.Phys., 12, 1--22 (2011)

    2011

  22. [22]

    Strominger, A., On BMS invariance of gravitational scattering, J.High Energy.Phys., 152 (2014)

    2014

  23. [23]

    High Energy.Phys., 86 (2016)

    Strominger, A., Zhiboedov, A., Gravitational memory, BMS supertranslations and soft theorems, J. High Energy.Phys., 86 (2016)

    2016

  24. [24]

    104, 207--226 (1986)

    Brown, J.D., Henneaux, M., Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity, Commum.Math.Phys. 104, 207--226 (1986)

    1986

  25. [25]

    Maldacena, J.M., The large n limit of superconformal field theories and supergravity, Int.J.Theor.Phys., 38, 1113--1133 (1999)

    1999

  26. [26]

    Pasterski, S., Shao, S.-H., Strominger, A., Flat space amplitudes and conformal symmetry of the celestial sphere, Phys.Rev.D, 96, 065026 (2017)

    2017

  27. [27]

    Hawking, S.W., Breakdown of predictability in gravitational collapse, Phys.Rev.D, 14, 2460 (1976)

    1976

  28. [28]

    Hawking, S.W., Perry, M.J., Strominger, A., Soft hair on black holes, Phys.Rev.Lett., 116, 231301 (2016)

    2016

  29. [29]

    Hawking, S.W., Perry, M.J., Strominger, A., Superrotation charge and supertranslation hair on black holes, J.High Energy.Phys., 161 (2017)

    2017

  30. [30]

    Di Francesco, P., Mathieu, P., Sénéchal, D., Conformal field theory, Graduate texts in contemporary physics, Springer, New York (1997)

    1997

  31. [31]

    Zuber, J.-B., Old and New Topics in Conformal Field Theory, Service de Physique Théorique de Saclay (1991)

    1991

  32. [32]

    Dickey, L.A., Lectures on Classical W-Algebras, Acta Applicandae Mathematicae, 47, 243--321 (1997)

    1997

  33. [33]

    Belavin, A.A., KdV-Type Equations and W-Algebras, Adv.Stud.Pure.Math, 19, 117--125 (1989)

    1989

  34. [34]

    Barakat, A., De Sole, A., Kac, V.G., Poisson vertex algebras in the theory of Hamiltonian equations, Jpn.J.Math., 4, 141–-252 (2009)

    2009

  35. [35]

    (eds) Perspectives in Lie Theory

    Kac, V., Introduction to Vertex Algebras, Poisson Vertex Algebras, and Integrable Hamiltonian PDE, In: Callegaro, F., Carnovale, G., Caselli, F., De Concini, C., De Sole, A. (eds) Perspectives in Lie Theory. Springer INdAM Series, vol 19, Springer, Cham (2017)

    2017

  36. [36]

    Pérez, A., Tempo, D., Troncoso, R., Boundary conditions for General Relativity on AdS _3 and the KdV hierarchy., J.High Energy.Phys., 103 (2016)

    2016

  37. [37]

    Oblak, B., BMS Particles in Three Dimensions, Springer Theses (2018)

    2018

  38. [38]

    Notes on the BMS group in three dimensions: II

    Barnich, G., Oblak, B. Notes on the BMS group in three dimensions: II. Coadjoint representation. J. High Energ. Phys. 33 (2015)

    2015

  39. [39]

    Notes on the BMS group in three dimensions: I

    Barnich, G., Oblak, B. Notes on the BMS group in three dimensions: I. Induced representations. J. High Energ. Phys. 129 (2014)

    2014

  40. [40]

    Rawnsley, J. H. Representations of a semi-direct product by quantization, Mathematical Proceedings of the Cambridge Philosophical Society 78,9, 345--350 (1975)

    1975

  41. [41]

    3--4, 245--270 (1998)

    Baguis, P., Semidirect products and the Pukanszky condition, Journal of Geometry and Physics,25, no. 3--4, 245--270 (1998)

    1998

  42. [42]

    Oblak, B., From the Lorentz group to the Celestial Sphere, arxiv: 1508.00920 (2018)

    Pith/arXiv arXiv 2018

  43. [43]

    Quantum Grav

    Barnich, G., Compère, G., Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quantum Grav. 24 3139 (2007)

    2007

  44. [44]

    Barnich, G., Troessaert, C., Supertranslations Call for Superrotations, Physics AUC, Vol.21-SP.Issue, 11--17 (2011)

    2011

  45. [45]

    Quantum Grav

    Prinz, D., Schmeding, A., Lie theory for asymptotic symmetries in general relativity: The BMS group, Class. Quantum Grav. 39 065004 (2022)

    2022

  46. [46]

    Ruzziconi, R., On the Various Extensions of the BMS Group, arxiv: 2009.01926, PhD Thesis (2020)

    Pith/arXiv arXiv 2009

  47. [47]

    Lie Theory 7,1, 61--99 (1997)

    Kriegl, A., Michor, P.W., Regular infinite dimensional Lie groups, J. Lie Theory 7,1, 61--99 (1997)

    1997

  48. [48]

    Kriegl, A., Michor, P.W., The convenient setting of global analysis, American Mathematical Soc. no. 53 (1997)

    1997

  49. [49]

    Guieu, L., Roger, C., L'algèbre et le groupe de Virasoro, Publications du CRM, Université de Montréal (2007)

    2007

  50. [50]

    The geometry of moments, Twistor Geometry and Non-Linear Systems

    Kirillov, A.A., Infinite dimensional lie groups; their orbits, invariants and representations. The geometry of moments, Twistor Geometry and Non-Linear Systems. Lecture Notes in Mathematics vol 970., Springer, Berlin, Heidelberg (1989)

    1989

  51. [51]

    Bakas, I., Conformal Invariance, the KdV equation and coadjoint orbits of the Virasoro algebra, Nuclear Physics B302, 189--203 (1988)

    1988

  52. [52]

    Gieres, F., Conformally Covariant Operators on Riemann Surfaces (with applications to conformal and integrable models) CERN.TH.5985/91, MPI-Ph/91-39 (1991)

    1991

  53. [53]

    12, 2961 (1995)

    Coussaert, O., Henneaux, M., van Driel, P., The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class.Quant.Grav. 12, 2961 (1995)

    1995

  54. [54]

    Barnich, G., Gomberoff, A., González, H.A., Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory, Phys. Rev. D 87, 124032 (2013)

    2013

  55. [55]

    Dorfman, I.Y., Dirac structures and Integrability of Nonlinear Evolution Equations, John Wiley & Sons Ltd, 188pp (1993)

    1993

  56. [56]

    Gel’fand, I.M., Dorfman, I.Y., Hamiltonian operators and algebraic structures related to them, Funk.Anal.Priloz., 13, 13--30 (1979)

    1979

  57. [57]

    Gel’fand, I.M., Dorfman, I.Y., Hamiltonian operators and infinite-dimensional Lie algebras, Funk.Anal.Priloz., 15, 173--187 (1981)

    1981

  58. [58]

    Dubrovin B., Zhang, Y., Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants, SISSA 65 (2001)

    2001

  59. [59]

    Falqui, G., Lorenzoni, P., Exact Poisson pencils, - structures and topological hierarchies, Physica D, 241, 2178--2187 (2012)

    2012

  60. [60]

    Fuchssteiner, B., Mastersymmetries, Higher Order Time-Dependent Symmetries and Conserved Densities of Nonlinear Evolution Equations, Progress of Theoretical Physics, Vol. 70, No. 6 (1983)

    1983

  61. [61]

    Zubelli, J.P., Magri, F., Differential Equations in the Spectral Parameter, Darboux Transformations and a Hierarchy of Master Symmetries for KdV, Commun.Math.Phys, 141, 329--351 (1991)

    1991

  62. [62]

    Duistermaat, J.J., Grünbaum, F.A., Differential Equations in the Spectral Parameter, Commun.Math.Phys., 103, 177--240 (1986)

    1986

  63. [63]

    Lenzi, M., Sopuerta, C.F., Darboux covariance: A hidden symmetry of perturbed Schwarzschild black holes, Phys.Rev.D, 104 (2021)

    2021

  64. [64]

    Jaramillo, J.L., Lenzi, M., Sopuerta, C.F., Integrability in perturbed black holes: Background hidden structures, Phys.Rev.D, 110 (2024)

    2024

  65. [65]

    Barnich, G., Gomberoff, A., González, H.A., Flat limit of three dimensional asymptotically anti–de Sitter spacetimes, Phys. Rev. D, 86 (2012)

    2012

  66. [66]

    Bagchi, A., Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories, Phys. Rev. Lett., 105 (2010)

    2010

  67. [67]

    Kirillov, A.A., Elements of the theory of representations, Grundlehren der mathematischen Wissenschaften GL, 220, Springer (1976)

    1976

  68. [68]

    Weinstein, A., The local structure of Poisson manifolds, J. Diff. Geom. 18, n. 3, 523--557 (1983)

    1983

  69. [69]

    162, 147--173 (1994)

    Alekseev, A.Yu., Malkin, A.Z., Symplectic Structures Associated to Lie-Poisson Groups, Commun.Math.Phys. 162, 147--173 (1994)

    1994

  70. [70]

    176, 116--144 (2003)

    Khesin, B.A., Misiolek G., Euler equations on homogeneous spaces and Virasoro orbits, Advances in Mathematics. 176, 116--144 (2003)

    2003

  71. [71]

    Arnold, V.I., Mathematical methods of classical mechanics, Graduate Texts in Mathematics, 60, Springer (1989)

    1989

  72. [72]

    71, 1661--1664 (1993)

    Camassa, R., Holm, D., An integrable shallow water equation with peaked solutions, Phys.Lett.Rev. 71, 1661--1664 (1993)

    1993

  73. [73]

    Hunter, J.K, Saxton, R., Dynamics of director fields, SIAM J. Appl. Math. 51 (6), 1498--1521 (1991)

    1991

  74. [74]

    21 (4), 81--82 (1987)

    Ovsienko, V., Khesin, B.A., The (super)KdV equation as an Euler equation, Funct.Anal.Appl. 21 (4), 81--82 (1987)

    1987

  75. [75]

    Misiolek, G., Conjugate Points in the Bott-Virasoro Group and the KdV Equation, Proc.American.Math.Soc., 125, 935--940 (1997)

    1997

  76. [76]

    Arnold, V.I., Khesin, B.A., Topological Methods in Hydrodynamics, Applied Math.Series 125, Springer (1998)

    1998

  77. [77]

    Adler, M., On a trace functional for formal pseudo-differential operators and the symplectic structure of the Korteweg-de Vries equations, Invent.Math 50, 219--248 (1979)

    1979

  78. [78]

    24, 81--180 (1984)

    Drinfeld, V.G., Sokolov, V.V., Lie algebras and equations of the Korteweg-de Vries type, Itogi Nauk Tekh., ser.Sovr.probl.mat. 24, 81--180 (1984)

    1984

  79. [79]

    86, 131--143 (1979)

    Wilson, G., Commuting flows and conservation laws for Lax equations, Math.Proc.Cambridge Phil.Soc. 86, 131--143 (1979)

    1979

  80. [80]

    Manin, Y.I., Algebraic aspects of nonlinear differential equations, J Math Sci 11, 1–122 (1979)

    1979

Showing first 80 references.