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

SIGMA 22 (2026), 070, 27 pages      arXiv:2604.08900      https://doi.org/10.3842/SIGMA.2026.070

Graded Casimir Elements and Central Extensions of Color Lie Algebras

Naruhiko Aizawa a, Ichi Fujii a, Jambulingam Segar b and Joris Van der Jeugt c
a) Department of Physics, Graduate School of Science, Osaka Metropolitan University, Sugimoto Campus, Osaka 558-8585, Japan
b) VICAS, Ramakrishna Mission Vivekananda College, Chennai-600 004, India
c) Department of Mathematics, Computer Science and Statistics, Ghent University,  Krijgslaan 281-S9, B-9000 Gent, Belgium

Received April 25, 2026, in final form July 30, 2026; Published online August 08, 2026

Abstract
A color Lie algebra is a generalization of a Lie (super)algebra by an Abelian group $\Gamma$. The underlying vector space and defining relations of the algebra are graded by $\Gamma$, and a color Lie algebra can admit graded Casimir elements. Furthermore, in that case its loop algebra admits graded central extensions. We present a general method for constructing 2nd order graded Casimir elements and graded central extensions for a given color Lie algebra and its loop algebra, respectively. We also show that there exists a large class of color Lie algebras admitting such graded Casimir elements or central extensions by providing three examples, namely, $\mathfrak{sl}(2)$ for $\Gamma = \mathbb{Z}_3^2$, and $\mathfrak{q}(n)$ and $\mathfrak{osp}(m|2n)$ for $\Gamma = \mathbb{Z}_2^2$.

Key words: color Lie algebra; Casimir element; central extension.

pdf (550 kb)   tex (39 kb)  

References

  1. Aizawa N., Amakawa K., Doi S., $\mathcal N$-extension of double-graded supersymmetric and superconformal quantum mechanics, J. Phys. A 53 (2020), 065205, 14 pages, arXiv:1905.06548.
  2. Aizawa N., Amakawa K., Doi S., $\mathbb{Z}_2^n$-graded extensions of supersymmetric quantum mechanics via Clifford algebras, J. Math. Phys. 61 (2020), 052105, 13 pages, arXiv:1912.11195.
  3. Aizawa N., Fujii I., Ito R., An integrable hierarchy associated with loop extension of $\mathbb Z_2^2$-graded $\mathfrak{osp}(1|2)$, J. Phys. A 59 (2026), 215206, 15 pages, arXiv:2512.14108.
  4. Aizawa N., Fujii I., Ito R., Tanaka T., Toppan F., Integrable $\mathbb Z_2^2$-graded super-Liouville equation and induced $\mathbb Z_2^2$-graded super-Virasoro algebra, J. Phys. A 59 (2026), 235202, 20 pages, arXiv:2512.17449.
  5. Aizawa N., Isaac P.S., Segar J., $\mathbb{Z}_2\times\mathbb{Z}_2$-generalizations of infinite-dimensional Lie superalgebra of conformal type with complete classification of central extensions, Rep. Math. Phys. 85 (2020), 351-373, arXiv:1902.05741.
  6. Aizawa N., Ito R., Integration on minimal $\mathbb Z^2_2$-superspace and emergence of space, J. Phys. A 56 (2023), 485201, 11 pages, arXiv:2305.07836.
  7. Aizawa N., Ito R., Kuznetsova Z., Tanaka T., Toppan F., Integrable $\mathbb Z^2_2$-graded extensions of the Liouville and Sinh-Gordon theories, J. Phys. A 58 (2025), 055201, 23 pages, arXiv:2406.13503.
  8. Aizawa N., Ito R., Tanaka T., $\mathbb Z^2_2$-graded supersymmetry via superfield on minimal $\mathbb Z^2_2$-superspace, J. Phys. A 57 (2024), 435201, 19 pages, arXiv:2308.16860.
  9. Aizawa N., Kimura D., Universal weight systems from a minimal $\mathbb Z_2^2$-graded Lie algebra, J. Knot Theory Ramifications 34 (2025), 2550045, 30 pages, arXiv:2410.05845.
  10. Aizawa N., Kuznetsova Z., Tanaka H., Toppan F., $\mathbb{Z}_2\times\mathbb{Z}_2$-graded Lie symmetries of the Lévy-Leblond equations, PTEP. Prog. Theor. Exp. Phys. 2016 (2016), 123A01, 26 pages, arXiv:1609.08224.
  11. Aizawa N., Kuznetsova Z., Tanaka H., Toppan F., Generalized supersymmetry and the Lévy-Leblond equation, in Physical and Mathematical Aspects of Symmetries, Springer, Cham, 2017, 79-84, arXiv:1609.08760.
  12. Aizawa N., Segar J., Affine extensions of $\mathbb Z_2^2$-graded $\mathfrak{osp}(1|2)$ and Virasoro algebra, Int. J. Geom. Methods Mod. Phys. 23 (2026), 2540052, 22 pages, arXiv:2409.07938.
  13. Alderete C., Green A.M., Nguyen N.H., Zhu Y., Linke N.M., Rodríguez-Lara B.M., Para-particle oscillator simulations on a trapped-ion quantum computer, J. Appl. Phys. 138 (2025), 054401, 8 pages, arXiv:2207.02430.
  14. Alderete C., Green A.M., Nguyen N.H., Zhu Y., Rodríguez-Lara B.M., Linke N.M., Experimental realization of para-particle oscillators, Sci. Rep. 15 (2025), 41498, 11 pages, arXiv:2108.05471.
  15. Alshammari F.S., Hoque M.F., Segar J., The $\mathfrak{osp}(1|2)$ $\mathbb{Z}_2 \times \mathbb{Z}_2$ graded algebra and its irreducible representations, arXiv:2306.12381.
  16. Amakawa K., Aizawa N., A classification of lowest weight irreducible modules over $\mathbb{Z}_2^2$-graded extension of $\mathfrak{osp}(1|2)$, J. Math. Phys. 62 (2021), 043502, 18 pages, arXiv:2011.03714.
  17. Babelon O., Bernard D., Talon M., Introduction to classical integrable systems, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 2003.
  18. Balbino M.M., de Freitas I.P., Rana R.G., Toppan F., Inequivalent $\mathbb{Z}_2^n$-graded brackets, $n$-bit parastatistics and statistical transmutations of supersymmetric quantum mechanics, Nuclear Phys. B 1009 (2024), 116729, 38 pages, arXiv:2309.00965.
  19. Bruce A.J., On a $\mathbb{Z}_2^n$-graded version of supersymmetry, Symmetry 11 (2019), 116, 20 pages, arXiv:1812.02943.
  20. Bruce A.J., $\mathbb Z_2 \times \mathbb Z_2$-graded supersymmetry: 2-d sigma models, J. Phys. A 53 (2020), 455201, 25 pages, arXiv:2006.08169.
  21. Bruce A.J., Is the $\mathbb Z_2\times\mathbb Z_2$-graded sine-Gordon equation integrable?, Nuclear Phys. B 971 (2021), 115514, 10 pages, arXiv:2106.06372.
  22. Bruce A.J., Duplij S., Double-graded supersymmetric quantum mechanics, J. Math. Phys. 61 (2020), 063503, 13 pages, arXiv:1904.06975.
  23. Bruce A.J., Ibarguengoytia E., Poncin N., The Schwarz-Voronov embedding of $\mathbb Z_2^n$-manifolds, SIGMA 16 (2020), 002, 47 pages, arXiv:1906.09834.
  24. Bruce A.J., Poncin N., Functional analytic issues in $\mathbb Z_2^n$-geometry, Rev. Un. Mat. Argentina 60 (2019), 611-636, arXiv:1807.11739.
  25. Bruce A.J., Poncin N., Products in the category of $\mathbb Z^n_2$-manifolds, J. Nonlinear Math. Phys. 26 (2019), 420-453, arXiv:1807.11740.
  26. Covolo T., Cohomological approach to the graded Berezinian, J. Noncommut. Geom. 9 (2015), 543-565, arXiv:1207.2962.
  27. Covolo T., Grabowski J., Poncin N., The category of $\mathbb{Z}_2^n$-supermanifolds, J. Math. Phys. 57 (2016), 073503, 16 pages, arXiv:1602.03312.
  28. Covolo T., Grabowski J., Poncin N., Splitting theorem for $\mathbb{Z}_2^n$-supermanifolds, J. Geom. Phys. 110 (2016), 393-401, arXiv:1602.03671.
  29. Covolo T., Kwok S., Poncin N., Local forms of morphisms of colored supermanifolds, J. Geom. Phys. 168 (2021), 104302, 21 pages, arXiv:2010.10026.
  30. Covolo T., Ovsienko V., Poncin N., Higher trace and Berezinian of matrices over a Clifford algebra, J. Geom. Phys. 62 (2012), 2294-2319, arXiv:1109.5877.
  31. Green H.S., Jarvis P.D., Casimir invariants, characteristic identities, and Young diagrams for color algebras and superalgebras, J. Math. Phys. 24 (1983), 1681-1687.
  32. Isaac P.S., Ryan M., The Dirac oscillator, generalised parastatistics and colour Lie superalgebras, Nuclear Phys. B 1020 (2025), 117154, 15 pages, arXiv:2507.14780.
  33. Isaac P.S., Stoilova N.I., Van der Jeugt J., The $\mathbb{Z}_2\times\mathbb{Z}_2$-graded general linear Lie superalgebra, J. Math. Phys. 61 (2020), 011702, 7 pages, arXiv:1912.08636.
  34. Ito R., Nago A., Novel possible symmetries of $S$-matrix generated by $\mathbb Z_2^n$-graded Lie superalgebras, Nuclear Phys. B 1014 (2025), 116877, 10 pages, arXiv:2501.07311.
  35. Jarvis P.D., Yang M., Wybourne B.G., Generalized quasispin for supergroups, J. Math. Phys. 28 (1987), 1192-1197.
  36. Kuznetsova Z., Toppan F., Classification of minimal $\mathbb Z_2\times\mathbb Z_2$-graded Lie (super)algebras and some applications, J. Math. Phys. 62 (2021), 063512, 22 pages, arXiv:2103.04385.
  37. Lu R., Tan Y., Construction of color Lie algebras from homomorphisms of modules of Lie algebras, J. Algebra 620 (2023), 1-49.
  38. Mohammadi M., Varsaie S., On the construction of $\mathbb Z^n_2$-grassmannians as homogeneous $\mathbb Z^n_2$-spaces, Electron. Res. Arch. 30 (2022), 221-241, arXiv:1901.07393.
  39. Piontkovski D., Silvestrov S.D., Cohomology of 3-dimensional color Lie algebras, J. Algebra 316 (2007), 499-513, arXiv:math.KT/0508573.
  40. Poncin N., Towards integration on colored supermanifolds, in Geometry of Jets and Fields, Banach Center Publ., Vol. 110, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 2016, 201-217.
  41. Poncin N., Schouten S., The geometry of supersymmetry: A concise introduction, Grad. J. Math. 8 (2023), 1-57, arXiv:2207.12974.
  42. Ree R., Generalized Lie elements, Canadian J. Math. 12 (1960), 493-502.
  43. Rittenberg V., Wyler D., Generalized superalgebras, Nuclear Phys. B 139 (1978), 189-202.
  44. Rittenberg V., Wyler D., Sequences of $Z_{2}\oplus Z_{2}$ graded Lie algebras and superalgebras, J. Math. Phys. 19 (1978), 2193-2200.
  45. Ryan M., Refining the grading of irreducible Lie colour algebra representations, arXiv:2403.02855.
  46. Ryan M., Graded colour Lie superalgebras for solving Lévy-Leblond equations, J. Phys. A 58 (2025), 015204, 24 pages, arXiv:2407.19723.
  47. Scheunert M., Generalized Lie algebras, J. Math. Phys. 20 (1979), 712-720.
  48. Scheunert M., Casimir elements of $\varepsilon $ Lie algebras, J. Math. Phys. 24 (1983), 2671-2680.
  49. Scheunert M., Graded tensor calculus, J. Math. Phys. 24 (1983), 2658-2670.
  50. Scheunert M., Zhang R.B., Cohomology of Lie superalgebras and their generalizations, J. Math. Phys. 39 (1998), 5024-5061, arXiv:q-alg/9701037.
  51. Sergeev A.N., The centre of enveloping algebra for Lie superalgebra $Q(n,\mathbb{C})$, Lett. Math. Phys. 7 (1983), 177-179.
  52. Stoilova N.I., Van der Jeugt J., The $\mathbb Z_2\times\mathbb Z_2$-graded Lie superalgebra $\mathfrak{pso}(2m+1|2n)$ and new parastatistics representations, J. Phys. A 51 (2018), 135201, 17 pages, arXiv:1711.02136.
  53. Stoilova N.I., Van der Jeugt J., On classical $\mathbb Z_2\times\mathbb Z_2$-graded Lie algebras, J. Math. Phys. 64 (2023), 061702, 8 pages, arXiv:2305.18604.
  54. Stoilova N.I., Van der Jeugt J., Orthosymplectic $\mathbb Z_2\times\mathbb Z_2$-graded Lie superalgebras and parastatistics, J. Phys. A 57 (2024), 095202, 13 pages, arXiv:2402.11952.
  55. Stoilova N.I., Van der Jeugt J., $\mathbb Z^3_2$-grading of the Lie algebra $G_2$ and related color algebras, J. Phys. A 58 (2025), 365201, 17 pages, arXiv:2505.04378.
  56. Tolstoy V.N., Once more on parastatistics, Phys. Part. Nucl. Lett. 11 (2014), 933-937, arXiv:1610.01628.
  57. Tolstoy V.N., Super-de Sitter and alternative super-Poincaré symmetries, in Lie Theory and Its Applications in Physics, Springer Proc. Math. Stat., Vol. 111, Springer, Tokyo, 2014, 357-367, arXiv:1610.01566.
  58. Toppan F., Inequivalent quantizations from gradings and $\mathbb{Z}_2\times\mathbb{Z}_2$ parabosons, J. Phys. A 54 (2021), 355202, 21 pages, arXiv:2104.09692.
  59. Toppan F., $\mathbb Z_2 \times \mathbb Z_2$-graded parastatistics in multiparticle quantum Hamiltonians, J. Phys. A 54 (2021), 115203, 35 pages, arXiv:2008.11554.
  60. Toppan F., First quantization of braided Majorana fermions, Nuclear Phys. B 980 (2022), 115834, 14 pages, arXiv:2203.01776.
  61. Toppan F., On braid statistics versus parastatistics, J. Phys. Conf. Ser. 2912 (2024), 012011, 11 pages, arXiv:2411.14261.
  62. Toppan F., On the detectability of paraparticles beyond bosons and fermions, Int. J. Geom. Methods Mod. Phys. 23 (2026), 2540042, 21 pages, arXiv:2411.18313.
  63. Vasiliev M.A., de Sitter supergravity with positive cosmological constant and generalised Lie superalgebras, Classical Quantum Gravity 2 (1985), 645-659.
  64. Wang Z., Hazzard K.R.A., Particle exchange statistics beyond fermions and bosons, Nature 637 (2025), 314-318, arXiv:2308.05203.
  65. Wills Toro L.A., $(I,q)$-graded Lie algebraic extensions of the Poincaré algebra, constraints on $I$ and $q$, J. Math. Phys. 36 (1995), 2085-2112.
  66. Wills-Toro L.A., Trefoil symmetries. I. Clover extensions beyond Coleman-Mandula theorem, J. Math. Phys. 42 (2001), 3915-3934.
  67. Zhang R.B., Invariants and representations of the $\Gamma$-graded general linear Lie $\omega$-algebras, Expo. Math. 44 (2026), 125778, 92 pages, arXiv:2509.21795.
  68. Zhang R.B., Quantum groups of Lie colour algebras fulfilling Cartan-Weyl paradigm, arXiv:2606.02202.

Previous article  Next article  Contents of Volume 22 (2026)