Kinematical superspaces

  • José Figueroa-O’FarrillEmail author
  • Ross Grassie
Open Access
Regular Article - Theoretical Physics


We classify N =1 d = 4 kinematical and aristotelian Lie superalgebras with spa- tial isotropy, but not necessarily parity nor time-reversal invariance. Employing a quater- nionic formalism which makes rotational covariance manifest and simplifies many of the calculations, we find a list of 43 isomorphism classes of Lie superalgebras, some with pa- rameters, whose (nontrivial) central extensions are also determined. We then classify their corresponding simply-connected homogeneous (4|4)-dimensional superspaces, resulting in a list of 27 homogeneous superspaces, some with parameters, all of which are reductive. We determine the invariants of low rank and explore how these superspaces are related via geometric limits.


Space-Time Symmetries Superspaces 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited


  1. [1]
    Yu. A. Golfand and E.P. Likhtman, Extension of the Algebra of Poincaŕe Group Generators and Violation of p Invariance, JETP Lett.13 (1971) 323 [INSPIRE].ADSGoogle Scholar
  2. [2]
    B. Zumino, Nonlinear Realization of Supersymmetry in de Sitter Space, Nucl. Phys.B 127 (1977) 189 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    P. de Medeiros, J. Figueroa-O’Farrill and A. Santi, Killing superalgebras for Lorentzian four-manifolds, JHEP06 (2016) 106 [arXiv:1605.00881] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    J. Figueroa-O’Farrill and S. Prohazka, Spatially isotropic homogeneous spacetimes, JHEP01 (2019) 229 [arXiv:1809.01224] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    J. Figueroa-O’Farrill, R. Grassie and S. Prohazka, Geometry and BMS Lie algebras of spatially isotropic homogeneous spacetimes, JHEP08 (2019) 119 [arXiv:1905.00034] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    H. Bacry and J. Levy-Leblond, Possible kinematics, J. Math. Phys.9 (1968) 1605 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  7. [7]
    H. Bacry and J. Nuyts, Classification of Ten-dimensional Kinematical Groups With Space Isotropy, J. Math. Phys.27 (1986) 2455 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    J. Rembielinski and W. Tybor, Possible superkinematics, Acta Phys. Polon.B 15 (1984) 611 [INSPIRE].MathSciNetGoogle Scholar
  9. [9]
    V. Hussin, J. Negro and M.A. del Olmo, Kinematical superalgebras, J. Phys.A 32 (1999) 5097.ADSMathSciNetzbMATHGoogle Scholar
  10. [10]
    R. Campoamor-Stursberg and M. Rausch de Traubenberg, Kinematical superalgebras and Lie algebras of order 3, J. Math. Phys.49 (2008) 063506 [arXiv:0801.2630] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    C.-G. Huang and L. Li, Possible Supersymmetric Kinematics, Chin. Phys.C 39 (2015) 093103 [arXiv:1409.5498] [INSPIRE].ADSCrossRefGoogle Scholar
  12. [12]
    R. Puzalowski, Galilean supersymmetry, Acta Phys. Austriaca50 (1978) 45 [INSPIRE].MathSciNetGoogle Scholar
  13. [13]
    F. Palumbo, Nonrelativistic Supersymmetry, in Proceedings of the International Conference on Recent Progress in Many Body Theories, International Center for Theoretical Physics, Trieste Italy (1978), pg. 582.Google Scholar
  14. [14]
    T.E. Clark and S.T. Love, Nonrelativistic supersymmetry, Nucl. Phys.B 231 (1984) 91 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    J.A. de Azcarraga and D. Ginestar, Nonrelativistic limit of supersymmetric theories, J. Math. Phys.32 (1991) 3500 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    J.M. Figueroa-O’Farrill, Kinematical Lie algebras via deformation theory, J. Math. Phys.59 (2018) 061701 [arXiv:1711.06111] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. Math.57 (1953) 591. [18] A. Santi, Superization of Homogeneous Spin Manifolds and Geometry of Homogeneous Supermanifolds, Abh. Math. Sem. Univ. Hamburg80 (2010) 87 [arXiv:0905.3832] [INSPIRE].
  18. [18]
    B. Kostant, Graded manifolds, graded Lie theory, and prequantization, in Conference on Differential Geometrical Methods in Mathematical Physics, Bonn Germany (1975), Lecture Notes Math.570 (1977) 177.CrossRefGoogle Scholar
  19. [19]
    M. Batchelor, The structure of supermanifolds, Trans. Am. Math. Soc.253 (1979) 329.MathSciNetCrossRefGoogle Scholar
  20. [20]
    J.-L. Koszul, Graded manifolds and graded Lie algebras, in Proceedings of the international meeting on geometry and physics, Florence Italy (1982), Pitagora, Bologna Italy (1983), pg. 71.Google Scholar
  21. [21]
    J. Matulich, S. Prohazka and J. Salzer, Limits of three-dimensional gravity and metric kinematical Lie algebras in any dimension, JHEP07 (2019) 118 [arXiv:1903.09165] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    J.M. Figueroa-O’Farrill, Conformal Lie algebras via deformation theory, J. Math. Phys.60 (2019) 021702 [arXiv:1809.03603] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    C. Duval and P.A. Horvathy, On Schr¨odinger superalgebras, J. Math. Phys.35 (1994) 2516 [hep-th/0508079] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    P. de Medeiros, J. Figueroa-O’Farrill and A. Santi, Killing superalgebras for Lorentzian six-manifolds, J. Geom. Phys.132 (2018) 13 [arXiv:1804.00319] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Maxwell Institute and School of MathematicsThe University of EdinburghEdinburghU.K.

Personalised recommendations