Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 085, 10 pages      arXiv:2102.04192      https://doi.org/10.3842/SIGMA.2026.085

On a Hidden Supersymmetry of Cosmological Billiards

Dimitry Leites a and Oleksandr Lozhechnyk b
a) Department of Mathematics, Stockholm University, Albanovägen 28, SE-114 19 Stockholm, Sweden
b) Faculty of Mechanics and Mathematics, Taras Shevchenko NationalUniversity of Kyiv, Hlushkova Avenue 4e, 03127 Kyiv, Ukraine

Received September 16, 2024, in final form August 21, 2026; Published online August 29, 2026

Abstract
In the Belinski-Khalatnikov-Lifshitz cosmological models with an oscillatory approximation to singularity, the tracks of cosmological billiards (whatever they are) are realized in the Weyl chambers of the hyperbolic Lie algebras. The latter were classified by Li Wang Lai; their super analogs — the almost affine Lie superalgebras — were classified in arXiv:0906.1860. Here, we observe that some of the 142 hyperbolic Lie algebras $H$ can be ''superized'' to almost affine Lie superalgebras $S(H)$: for each of these $H$, it is possible to divide a row of its Cartan matrix by 2, simultaneously changing the parity of the corresponding Chevalley generators and relations between them. For the $H$ with a symmetrizable Cartan matrix, we list all 97 such pairs ($H\longleftrightarrow S(H)$). Several (18 of the total 66) thus ''superizable'' algebras $H$ have multiple such ''superizations''. The tracks of cosmological billiards corresponding to both terms of these pairs ($H\longleftrightarrow S(H)$) coincide since $H$ and $S(H)$ have one and the same Weyl chamber. We also classify ''superizations'' of hyperbolic Lie algebras with non-symmetrizable Cartan matrix, and pairs $\mathfrak{g}\longleftrightarrow \mathfrak{g}_{\bar 0}$, where $\mathfrak{g}$ is a simple finite-dimensional Lie superalgebra without Cartan matrix and simple component $\mathfrak{g}_{\bar 0}$.

Key words: hyperbolic Lie algebra; almost affine Lie superalgebra; Cartan matrix; cosmological billiard.

pdf (388 kb)   tex (24 kb)  

References

  1. Belinski V., Henneaux M., The cosmological singularity, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 2018.
  2. Bouarroudj S., Grozman P., Lebedev A., Leites D., Shchepochkina I., Simple vectorial Lie algebras in characteristic 2 and their superizations, SIGMA 16 (2020), 089, 101 pages, arXiv:1510.07255.
  3. Bouarroudj S., Grozman P., Leites D., Classification of finite dimensional modular Lie superalgebras with indecomposable Cartan matrix, SIGMA 5 (2009), 060, 63 pages, arXiv:0710.5149.
  4. Bouarroudj S., Grozman P., Leites D., Defining relations of almost affine (hyperbolic) Lie superalgebras, J. Nonlinear Math. Phys. 17 (2010), suppl. 1, 163-168, arXiv:1012.0176.
  5. Bouarroudj S., Krutov A., Leites D., Shchepochkina I., Non-degenerate invariant (super)symmetric bilinear forms on simple Lie (super)algebras, Algebr. Represent. Theory 21 (2018), 897-941, arXiv:1806.05505.
  6. Bouarroudj S., Lebedev A., Leites D., Shchepochkina I., Classification of simple Lie superalgebras in characteristic 2, Int. Math. Res. Not. 2023 (2023), 54-94, arXiv:1407.1695.
  7. Bouarroudj S., Leites D., Shang J., Computer-aided study of double extensions of restricted lie superalgebras preserving the nondegenerate closed 2-forms in characteristic 2, Exp. Math. 31 (2022), 676-688, arXiv:1904.09579.
  8. Carbone L., Chung S., Cobbs L., McRae R., Nandi D., Naqvi Y., Penta D., Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits, J. Phys. A 43 (2010), 155209, 30, arXiv:1003.0564.
  9. Chapovalov D., Chapovalov M., Lebedev A., Leites D., The classification of almost affine (hyperbolic) Lie superalgebras, J. Nonlinear Math. Phys. 17 (2010), 103-161, arXiv:0906.1860.
  10. Damour T., Henneaux M., Chaos in superstring cosmology, Phys. Rev. Lett. 85 (2000), 920-923, arXiv:hep-th/0003139.
  11. Damour T., Henneaux M., Julia B., Nicolai H., Hyperbolic Kac-Moody algebras and chaos in Kaluza-Klein models, Phys. Lett. B 509 (2001), 323-330, arXiv:hep-th/0103094.
  12. Damour T., Henneaux M., Nicolai H., Cosmological billiards, Classical Quantum Gravity 20 (2003), R145-R200, arXiv:hep-th/0212256.
  13. Damour T., Spindel P., Quantum supersymmetric cosmological billiards and their hidden Kac-Moody structure, Phys. Rev. D 95 (2017), 126011, 23 pages, arXiv:1704.08116.
  14. de Buyl S., Schomblond C., Hyperbolic Kac Moody algebras and Einstein billiards, J. Math. Phys. 45 (2004), 4464-4492, arXiv:hep-th/0403285.
  15. Deligne P., Lehrer G.I., Zhang R.B., The first fundamental theorem of invariant theory for the orthosymplectic super group, Adv. Math. 327 (2018), 4-24, arXiv:1508.04202.
  16. Dynkin diagram, Wikipedia, https://en.wikipedia.org/wiki/Dynkin_diagram.
  17. Grozman P., Leites D., Defining relations for Lie superalgebras with Cartan matrix, Czechoslovak J. Phys. 51 (2001), 1-21, arXiv:hep-th/9702073.
  18. Hoyt C., Serganova V., Classification of finite-growth general Kac-Moody superalgebras, Comm. Algebra 35 (2007), 851-874, arXiv:0810.2637.
  19. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  20. Khalatnikov I.M., Kamenshchik A.Yu., Lev Landau and the problem of singularities in cosmology, Phys. Uspekhi 51 (2008), 609-616, arXiv:0803.2684.
  21. Kleinschmidt A., Koehn M., Nicolai H., Supersymmetric quantum cosmological billiards, Phys. Rev. D 80 (2009), 061701, 5 pages, arXiv:0907.3048.
  22. Kobayashi Z., Morita J., Automorphisms of certain root lattices, Tsukuba J. Math. 7 (1983), 323-336.
  23. Leites D., Saveliev M.V., Serganova V.V., Embeddings of $\mathfrak{osp}(N|2)$ and completely integrable systems, in Group-Theoretical Methods in Physics (Yurmala, May 1985), Nauka, Moscow, 1987, 377-394 (an enlarged version in English is published by VNU Sci Press, 1986, 255-297).
  24. Leites D., Shchepochkina I., The classification of simple complex Lie superalgebras of polynomial vector fields and their deformations, arXiv:2406.16760.
  25. Li W.L., Classification of generalized Cartan matrices of hyperbolic type, Chinese Ann. Math. Ser. B 9 (1988), 68-77.
  26. Lifshitz E.M., Khalatnikov I.M., Investigations in relativistic cosmology, Adv. Phys. 12 (1963), 185-249.
  27. Lozhechnyk O., Hyperbolic superizations, https://github.com/sash-repo/hyperbolic-superizations.
  28. Mathieu O., Classification of simple graded Lie algebras of finite growth, Invent. Math. 108 (1992), 455-519.
  29. Saçliouglu C., Dynkin diagrams for hyperbolic Kac-Moody algebras, J. Phys. A 22 (1989), 3753-3769.
  30. Serganova V.V., Automorphisms of simple Lie superalgebras, Math. USSR-Izv. 24 (1985), 539-551.
  31. van de Leur J.W., A classification of contragredient Lie superalgebras of finite growth, Comm. Algebra 17 (1989), 1815-1841.

Previous article  Next article  Contents of Volume 22 (2026)