On rigidity of 3d asymptotic symmetry algebras

  • A. Farahmand Parsa
  • H. R. SafariEmail author
  • M. M. Sheikh-Jabbari
Open Access
Regular Article - Theoretical Physics


We study rigidity and stability of infinite dimensional algebras which are not subject to the Hochschild-Serre factorization theorem. In particular, we consider algebras appearing as asymptotic symmetries of three dimensional spacetimes, the \( \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_3 \), \( \mathfrak{u}(1) \) Kac-Moody and Virasoro algebras. We construct and classify the family of algebras which appear as deformations of \( \mathfrak{b}\mathfrak{m}{\mathfrak{s}}_3 \), \( \mathfrak{u}(1) \) Kac-Moody and their central extensions by direct computations and also by cohomological analysis. The Virasoro algebra appears as a specific member in this family of rigid algebras; for this case stabilization procedure is inverse of the Inönü-Wigner contraction relating Virasoro to bms3 algebra. We comment on the physical meaning of deformation and stabilization of these algebras and relevance of the family of rigid algebras we obtain.


Conformal and W Symmetry Differential and Algebraic Geometry Gaugegravity correspondence Space-Time Symmetries 


Open Access

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  1. [1]
    M. Levy-Nahas, Deformation and contraction of Lie algebras, J. Math. Phys. 8 (1967) 1211.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M. Levy-Nahas and R. Seneor, First order deformations of Lie algebra representations, e(3) and Poincaré examples, Commun. Math. Phys. 9 (1968) 242.ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    J.M. Figueroa-O’Farrill, Deformations of the Galilean Algebra, J. Math. Phys. 30 (1989) 2735 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    R. Vilela Mendes, Deformations, stable theories and fundamental constants, J. Phys. A 27 (1994) 8091 [INSPIRE].
  5. [5]
    C. Chryssomalakos and E. Okon, Generalized quantum relativistic kinematics: A Stability point of view, Int. J. Mod. Phys. D 13 (2004) 2003 [hep-th/0410212] [INSPIRE].
  6. [6]
    J. Figueroa-O’Farrill, Classification of kinematical Lie algebras, arXiv:1711.05676 [INSPIRE].
  7. [7]
    J.M. Figueroa-O’Farrill, Kinematical Lie algebras via deformation theory, J. Math. Phys. 59 (2018) 061701 [arXiv:1711.06111] [INSPIRE].
  8. [8]
    J.M. Figueroa-O’Farrill, Higher-dimensional kinematical Lie algebras via deformation theory, J. Math. Phys. 59 (2018) 061702 [arXiv:1711.07363] [INSPIRE].
  9. [9]
    T. Andrzejewski and J.M. Figueroa-O’Farrill, Kinematical lie algebras in 2 + 1 dimensions, J. Math. Phys. 59 (2018) 061703 [arXiv:1802.04048] [INSPIRE].
  10. [10]
    J.M. Figueroa-O’Farrill, Conformal Lie algebras via deformation theory, arXiv:1809.03603 [INSPIRE].
  11. [11]
    J. Figueroa-O’Farrill and S. Prohazka, Spatially isotropic homogeneous spacetimes, JHEP 01 (2019) 229 [arXiv:1809.01224] [INSPIRE].
  12. [12]
    E. Inonu and E.P. Wigner, On the Contraction of groups and their represenations, Proc. Nat. Acad. Sci. 39 (1953) 510 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    J.-M. Lévy-Leblond, Une nouvelle limite non-relativiste du groupe de Poincaré, Ann. Inst. H. Poincaré 3 (1965) 1.MathSciNetzbMATHGoogle Scholar
  14. [14]
    R. Gilmore, Lie groups, Lie algebras, and some of their applications, Courier Corporation, North Chelmsford U.S.A. (2012).Google Scholar
  15. [15]
    A. Fialowski and M. Penkava, Deformation theory of infinity algebras, J. Algebra 255 (2002) 59 [math/0101097].
  16. [16]
    A. Fialowski, Formal rigidity of the Witt and Virasoro algebra, J. Math. Phys. 53 (2012) 073501.Google Scholar
  17. [17]
    S. Gao, C. Jiang and Y. Pei, The derivations, central extensions and automorphism group of the Lie algebra W, arXiv:0801.3911.
  18. [18]
    S. Gao, C. Jiang and Y. Pei, Low-dimensional cohomology groups of the Lie algebras W(a, b), Commun. Algebra 39 (2011) 397.Google Scholar
  19. [19]
    J. Ecker and M. Schlichenmaier, The Vanishing of the Low-Dimensional Cohomology of the Witt and the Virasoro algebra, arXiv:1707.06106 [INSPIRE].
  20. [20]
    J. Ecker and M. Schlichenmaier, The Low-Dimensional Algebraic Cohomology of the Virasoro Algebra, arXiv:1805.08433 [INSPIRE].
  21. [21]
    J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
  22. [22]
    G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
  23. [23]
    A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
  24. [24]
    A. Campoleoni, D. Francia and C. Heissenberg, Asymptotic Charges at Null Infinity in Any Dimension, Universe 4 (2018) 47 [arXiv:1712.09591] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    G. Barnich, Centrally extended BMS4 Lie algebroid, JHEP 06 (2017) 007 [arXiv:1703.08704] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    H. Afshar, E. Esmaeili and M.M. Sheikh-Jabbari, Asymptotic Symmetries in p-Form Theories, JHEP 05 (2018) 042 [arXiv:1801.07752] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    P. Concha, N. Merino, O. Mišković, E. Rodríguez, P. Salgado-ReboLledó and O. Valdivia, Asymptotic symmetries of three-dimensional Chern-Simons gravity for the Maxwell algebra, JHEP 10 (2018) 079 [arXiv:1805.08834] [INSPIRE].
  28. [28]
    V. Hosseinzadeh, A. Seraj and M.M. Sheikh-Jabbari, Soft Charges and Electric-Magnetic Duality, JHEP 08 (2018) 102 [arXiv:1806.01901] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    A. Seraj, Conserved charges, surface degrees of freedom and black hole entropy, Ph.D. Thesis, Institute for Research in Fundamental Sciences, Tehran Iran (2016) [arXiv:1603.02442] [INSPIRE].
  30. [30]
    G. Compère and A. Fiorucci, Advanced Lectures on General Relativity, Lect. Notes Phys. 952 (2019) pp. [arXiv:1801.07064] [INSPIRE].
  31. [31]
    H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
  32. [32]
    R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
  34. [34]
    G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS(CNCFG2010)010 [arXiv:1102.4632] [INSPIRE].
  35. [35]
    A. Ashtekar, J. Bičák and B.G. Schmidt, Asymptotic structure of symmetry reduced general relativity, Phys. Rev. D 55 (1997) 669 [gr-qc/9608042] [INSPIRE].
  36. [36]
    G. Barnich and G. Compere, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [gr-qc/0610130] [INSPIRE].
  37. [37]
    B. Oblak, BMS Particles in Three Dimensions, Ph.D. Thesis, Brussels University, Brussels Belgium (2016) [arXiv:1610.08526] [INSPIRE].
  38. [38]
    G. Compère and S. Detournay, Boundary conditions for spacelike and timelike warped AdS 3 spaces in topologically massive gravity, JHEP 08 (2009) 092 [arXiv:0906.1243] [INSPIRE].
  39. [39]
    G. Compère, M. Guica and M.J. Rodriguez, Two Virasoro symmetries in stringy warped AdS 3, JHEP 12 (2014) 012 [arXiv:1407.7871] [INSPIRE].
  40. [40]
    G. Compère, L. Donnay, P.-H. Lambert and W. Schulgin, Liouville theory beyond the cosmological horizon, JHEP 03 (2015) 158 [arXiv:1411.7873] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    C. Troessaert, Enhanced asymptotic symmetry algebra of AdS 3, JHEP 08 (2013) 044 [arXiv:1303.3296] [INSPIRE].
  42. [42]
    H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].
  43. [43]
    D. Grumiller and M. Riegler, Most general AdS 3 boundary conditions, JHEP 10 (2016) 023 [arXiv:1608.01308] [INSPIRE].
  44. [44]
    D. Grumiller, W. Merbis and M. Riegler, Most general flat space boundary conditions in three-dimensional Einstein gravity, Class. Quant. Grav. 34 (2017) 184001 [arXiv:1704.07419] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    H. Afshar, D. Grumiller and M.M. Sheikh-Jabbari, Near horizon soft hair as microstates of three dimensional black holes, Phys. Rev. D 96 (2017) 084032 [arXiv:1607.00009] [INSPIRE].
  46. [46]
    H. Afshar, D. Grumiller, M.M. Sheikh-Jabbari and H. Yavartanoo, Horizon fluff, semi-classical black hole microstatesLog-corrections to BTZ entropy and black hole/particle correspondence, JHEP 08 (2017) 087 [arXiv:1705.06257] [INSPIRE].
  47. [47]
    D. Grumiller, A. Perez, S. Prohazka, D. Tempo and R. Troncoso, Higher Spin Black Holes with Soft Hair, JHEP 10 (2016) 119 [arXiv:1607.05360] [INSPIRE].
  48. [48]
    A. Campoleoni, D. Francia and C. Heissenberg, On higher-spin supertranslations and superrotations, JHEP 05 (2017) 120 [arXiv:1703.01351] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    A. Fialowski, Deformations of some infinite-dimensional Lie algebras, J. Math. Phys. 31 (1990) 1340.Google Scholar
  50. [50]
    A. Fialowski and M. Schlichenmaier, Global deformations of the Witt algebra of Krichever-Novikov type, Commun. Contemp. Math. 5 (2003) 921.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    M. Schlichenmaier, An elementary proof of the vanishing of the second cohomology of the Witt and Virasoro algebra with values in the adjoint module, Forum Math. 26 (2014) 913.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    G. Barnich, A. Gomberoff and H.A. González, Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory, Phys. Rev. D 87 (2013) 124032 [arXiv:1210.0731] [INSPIRE].
  53. [53]
    I.M. Gel’fand and D. Fuks, Cohomologies of Lie algebra of tangential vector fields of a smooth manifold, Funct. Anal. Appl. 3 (1969) 194.CrossRefzbMATHGoogle Scholar
  54. [54]
    G. Compère, W. Song and A. Strominger, New Boundary Conditions for AdS3, JHEP 05 (2013) 152 [arXiv:1303.2662] [INSPIRE].
  55. [55]
    W. Li, W. Song and A. Strominger, Chiral Gravity in Three Dimensions, JHEP 04 (2008) 082 [arXiv:0801.4566] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    M. Bañados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time, Phys. Rev. Lett. 69 (1992) 1849 [hep-th/9204099] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    H. Afshar, D. Grumiller, W. Merbis, A. Perez, D. Tempo and R. Troncoso, Soft hairy horizons in three spacetime dimensions, Phys. Rev. D 95 (2017) 106005 [arXiv:1611.09783] [INSPIRE].
  58. [58]
    M. Gerstenhaber, On the deformation of rings and algebras: I, Ann. Math. 79 (1964) 59.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [59]
    M. Gerstenhaber, On the deformation of rings and algebras: II, Ann. Math. 84 (1966) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    M. Gerstenhaber, On the deformation of rings and algebras: III, Ann. Math. 88 (1968) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    M. Gerstenhaber, On the deformation of rings and algebras: IV, Ann. Math. 99 (1974) 257.MathSciNetCrossRefzbMATHGoogle Scholar
  62. [62]
    A. Nijenhuis and R. Richardson, Deformations of Lie algebra structures, J. Math. Mech. 17 (1967) 89.MathSciNetzbMATHGoogle Scholar
  63. [63]
    A. Fialowski, Deformations of Lie algebras, Math. USSR Sb. 55 (1986) 467.CrossRefGoogle Scholar
  64. [64]
    A. Fialowski, An example of formal deformations of Lie algebras, in Deformation theory of algebras and structures and applications, Springer, Berlin Germany (1988) pg. 375.Google Scholar
  65. [65]
    A. Fialowski and M. Schlichenmaier, Global geometric deformations of current algebras as Krichever-Novikov type algebras, Commun. Math. Phys. 260 (2005) 579 [math/0412113] [INSPIRE].
  66. [66]
    L. Guerrini, Formal and analytic deformations of the Witt algebra, Lett. Math. Phys. 46 (1998) 121.Google Scholar
  67. [67]
    L. Guerrini, Formal and analytic rigidity of the Witt algebra, Rev. Math. Phys. 11 (1999) 303.Google Scholar
  68. [68]
    A. Onishchik and E.B. Vinberg, Encyclopaedia of Mathematical Sciences. Vol 41: Lie groups and Lie algebras III, structure of Lie groups and Lie algebras, Springer, Heidelberg Germany (1994).Google Scholar
  69. [69]
    D.B. Fuks, Cohomology of infinite-dimensional Lie algebras, Springer Science & Business Media, Berlin Germany (2012).Google Scholar
  70. [70]
    C. Chevalley and S. Eilenberg, Cohomology Theory of Lie Groups and Lie Algebras, Trans. Am. Math. Soc. 63 (1948) 85 [INSPIRE].
  71. [71]
    G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. Math. 57 (1953) 59.CrossRefzbMATHGoogle Scholar
  72. [72]
    C. Roger and J. Unterberger, The Schrödinger-Virasoro Lie group and algebra: From geometry to representation theory, Annales Henri Poincaré 7 (2006) 1477 [math-ph/0601050] [INSPIRE].
  73. [73]
    A. Nijenhuis and R.W. Richardson, Jr., Cohomology and deformations in graded Lie algebras, Bull. Am. Math. Soc. 72 (1966) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [74]
    M. Goze, Lie algebras: Classification, deformations and rigidity, math/0611793.
  75. [75]
    R. Richardson, On the rigidity of semi-direct products of Lie algebras, Pac. J. Math. 22 (1967) 339.Google Scholar
  76. [76]
    A. Fialowski and M. de Montigny, Deformations and contractions of Lie algebras, J. Phys. A 38 (2005) 6335.Google Scholar
  77. [77]
    I.E. Segal et al., A class of operator algebras which are determined by groups, Duke Math. J. 18 (1951) 221.Google Scholar
  78. [78]
    J. Patera, Graded contractions of Lie algebras, representations and tensor products, AIP Conf. Proc. 266 (1992) 46 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  79. [79]
    E. Weimar-Woods, Contractions of Lie algebras: generalized Inönü-Wigner contractions versus graded contractions, J. Math. Phys. 36 (1995) 4519.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  80. [80]
    D. Degrijse and N. Petrosyan, On cohomology of split Lie algebra extensions, J. Lie Theory 22 (2012) 1 [arXiv:0911.0545].
  81. [81]
    M. Hazewinkel and M. Gerstenhaber, Deformation theory of algebras and structures and applications. Vol. 247, Springer Science & Business Media, Amsterdam The Netherlands (2012).Google Scholar
  82. [82]
    P. Christe and M. Henkel, Introduction to Conformal Invariance and its Applications to Critical Phenomena, Lect. Notes Phys. Monogr. 16 (1993) 1 [cond-mat/9304035] [INSPIRE].
  83. [83]
    M. Henkel, A short introduction to conformal invariance, Lect. Notes Phys. 853 (2012) 1 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  84. [84]
    J. Unterberger and C. Roger, The Schrödinger-Virasoro Algebra: Mathematical Structure and Dynamical Schrödinger Symmetries, Springer Science & Business Media, Berlin Germany (2011).Google Scholar
  85. [85]
    P. Majumdar, Inönü-Wigner contraction of Kac-Moody algebras, J. Math. Phys. 34 (1993) 2059 [hep-th/9207057] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  86. [86]
    C. Daboul, J. Daboul and M. de Montigny, Gradings and contractions of affine Kac-Moody algebras, J. Math. Phys. 49 (2008) 063509.Google Scholar
  87. [87]
    V.Y. Ovsienko and C. Roger, Extensions of the Virasoro group and the Virasoro algebra by modules of tensor densities on S, Funct. Anal. Appl. 30 (1996) 290.Google Scholar
  88. [88]
    A. Bagchi, S. Detournay, R. Fareghbal and J. Simòn, Holography of 3D Flat Cosmological Horizons, Phys. Rev. Lett. 110 (2013) 141302 [arXiv:1208.4372] [INSPIRE].ADSCrossRefGoogle Scholar
  89. [89]
    G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  90. [90]
    C. Troessaert, The BMS4 algebra at spatial infinity, Class. Quant. Grav. 35 (2018) 074003 [arXiv:1704.06223] [INSPIRE].
  91. [91]
    P.J. McCarthy, Lifting of projective representations of the Bondi-Metzner-Sachs group, Proc. Roy. Soc. Lond. A 358 (1978) 141.Google Scholar
  92. [92]
    G. Compère, K. Hajian, A. Seraj and M.M. Sheikh-Jabbari, Extremal Rotating Black Holes in the Near-Horizon Limit: Phase Space and Symmetry Algebra, Phys. Lett. B 749 (2015) 443 [arXiv:1503.07861] [INSPIRE].
  93. [93]
    G. Compère, K. Hajian, A. Seraj and M.M. Sheikh-Jabbari, Wiggling Throat of Extremal Black Holes, JHEP 10 (2015) 093 [arXiv:1506.07181] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  94. [94]
    R. Javadinezhad, B. Oblak and M.M. Sheikh-Jabbari, Near-horizon extremal geometries: coadjoint orbits and quantization, JHEP 04 (2018) 025 [arXiv:1712.07627] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar

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© The Author(s) 2019

Authors and Affiliations

  1. 1.School of Mathematics, Institute for Research in Fundamental Sciences (IPM)TehranIran
  2. 2.School of Mathematics, TATA Institute of Fundamental Research (TIFR)MumbaiIndia
  3. 3.School of Physics, Institute for Research in Fundamental Sciences (IPM)TehranIran

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