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

SIGMA 15 (2019), 082, 16 pages      arXiv:1904.03993

One Parameter Family of Jordanian Twists

Daniel Meljanac a, Stjepan Meljanac b, Zoran Škoda c and Rina Štrajn d
a) Division of Materials Physics, Institute Rudjer Bošković, Bijenička cesta 54, P.O. Box 180, HR-10002 Zagreb, Croatia
b) Theoretical Physics Division, Institute Rudjer Bošković, Bijenička cesta 54, P.O. Box 180, HR-10002 Zagreb, Croatia
c) Department of Teachers' Education, University of Zadar, Franje Tudjmana 24, 23000 Zadar, Croatia
d) Department of Electrical Engineering and Computing, University of Dubrovnik, Ćira Carića 4, 20000 Dubrovnik, Croatia

Received April 16, 2019, in final form October 19, 2019; Published online October 25, 2019

We propose an explicit generalization of the Jordanian twist proposed in $r$-symmetrized form by Giaquinto and Zhang. It is proved that this generalization satisfies the 2-cocycle condition. We present explicit formulas for the corresponding star product and twisted coproduct. Finally, we show that our generalization coincides with the twist obtained from the simple Jordanian twist by twisting by a 1-cochain.

Key words: noncommutative geometry; Jordanian twist.

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  1. Borowiec A., Meljanac D., Meljanac S., Pachoł A., Interpolations between Jordanian twists induced by coboundary twists, SIGMA 15 (2019), 054, 22 pages, arXiv:1812.05535.
  2. Borowiec A., Pachoł A., $\kappa$-Minkowski spacetime as the result of Jordanian twist deformation, Phys. Rev. D 79 (2009), 045012, 11 pages, arXiv:0812.0576.
  3. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1994.
  4. Coll V., Gerstenhaber M., Giaquinto A., An explicit deformation formula with noncommuting derivations, in Ring Theory 1989 (Ramat Gan and Jerusalem, 1988/1989), Israel Math. Conf. Proc., Vol. 1, Weizmann, Jerusalem, 1989, 396-403.
  5. Demidov E.E., Manin Yu.I., Mukhin E.E., Zhdanovich D.V., Nonstandard quantum deformations of ${\rm GL}(n)$ and constant solutions of the Yang-Baxter equation, Progr. Theoret. Phys. Suppl. 102 (1990), 203-218.
  6. Dimitrijević M., Jonke L., Pachoł A., Gauge theory on twisted $\kappa$-Minkowski: old problems and possible solutions, SIGMA 10 (2014), 063, 22 pages, arXiv:1403.1857.
  7. Drinfel'd V.G., Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254-258.
  8. Etingof P., Kazhdan D., Quantization of Lie bialgebras. I, Selecta Math. (N.S.) 2 (1996), 1-41, arXiv:q-alg/9506005.
  9. Etingof P., Schiffmann O., Lectures on quantum groups, 2nd ed., Lectures in Math. Phys., International Press, Somerville, MA, 2002.
  10. Giaquinto A., Zhang J.J., Bialgebra actions, twists, and universal deformation formulas, J. Pure Appl. Algebra 128 (1998), 133-151, arXiv:hep-th/9411140.
  11. Govindarajan T.R., Gupta K.S., Harikumar E., Meljanac S., Meljanac D., Twisted statistics in $\kappa$-Minkowski spacetime, Phys. Rev. D 77 (2008), 105010, 6 pages, arXiv:0802.1576.
  12. Gräbe H.-G., Vlassov A.T., On a formula of Coll-Gerstenhaber-Giaquinto, J. Geom. Phys. 28 (1998), 129-142.
  13. Hoare B., van Tongeren S.J., On jordanian deformations of ${\rm AdS}_5$ and supergravity, J. Phys. A: Math. Theor. 49 (2016), 434006, 22 pages, arXiv:1605.03554.
  14. Jurić T., Kovačević D., Meljanac S., $\kappa$-deformed phase space, Hopf algebroid and twisting, SIGMA 10 (2014), 106, 18 pages, arXiv:1402.0397.
  15. Jurić T., Meljanac S., Pikutić D., Realizations of $\kappa$-Minkowski space, Drinfeld twists and related symmetry algebra, Eur. Phys. J. C Part. Fields 75 (2015), 528, 16 pages, arXiv:1506.04955.
  16. Jurić T., Meljanac S., Pikutić D., Families of vector-like deformations of relativistic quantum phase spaces, twists and symmetries, Eur. Phys. J. C Part. Fields 77 (2017), 830, 12 pages, arXiv:1709.04745.
  17. Jurić T., Meljanac S., Štrajn R., $\kappa$-Poincaré-Hopf algebra and Hopf algebroid structure of phase space from twist, Phys. Lett. A 377 (2013), 2472-2476, arXiv:1303.0994.
  18. Jurić T., Meljanac S., Štrajn R., Twists, realizations and Hopf algebroid structure of $\kappa$-deformed phase space, Internat. J. Modern Phys. A 29 (2014), 1450022, 32 pages, arXiv:1305.3088.
  19. Khoroshkin S.M., Pop I.I., Samsonov M.E., Stolin A.A., Tolstoy V.N., On some Lie bialgebra structures on polynomial algebras and their quantization, Comm. Math. Phys. 282 (2008), 625-662, arXiv:0706.1651.
  20. Khoroshkin S.M., Stolin A.A., Tolstoy V.N., Deformation of Yangian $Y({\rm sl}_2)$, Comm. Algebra 26 (1998), 1041-1055, arXiv:q-alg/9511005.
  21. Khoroshkin S.M., Stolin A.A., Tolstoy V.N., $q$-power function over $q$-commuting variables and deformed $XXX$ and $XXZ$ chains, Phys. Atomic Nuclei 64 (2001), 2173-2178, arXiv:math.QA/0012207.
  22. Kovačević D., Meljanac S., Pachoł A., Štrajn R., Generalized Poincaré algebras, Hopf algebras and $\kappa$-Minkowski spacetime, Phys. Lett. B 711 (2012), 122-127, arXiv:1202.3305.
  23. Kulish P.P., Stolin A.A., Deformed Yangians and integrable models, Czechoslovak J. Phys. 47 (1997), 1207-1212, arXiv:q-alg/9708024.
  24. Lukierski J., Nowicki A., Ruegg H., New quantum Poincaré algebra and $\kappa$-deformed field theory, Phys. Lett. B 293 (1992), 344-352.
  25. Lukierski J., Ruegg H., Nowicki A., Tolstoy V.N., $q$-deformation of Poincaré algebra, Phys. Lett. B 264 (1991), 331-338.
  26. Lyubashenko V.V., Hopf algebras and vector-symmetries, Russian Math. Surveys 41 (1986), no. 5, 153-154.
  27. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  28. Meljanac D., Meljanac S., Mignemi S., Štrajn R., $\kappa$-deformed phase spaces, Jordanian twists, Lorentz-Weyl algebra, Phys. Rev. D 99 (2019), 126012, 12 pages, arXiv:1903.08679.
  29. Meljanac S., Krešić-Jurić S., Differential structure on $\kappa$-Minkowski space, and $\kappa$-Poincaré algebra, Internat. J. Modern Phys. A 26 (2011), 3385-3402, arXiv:1004.4647.
  30. Meljanac S., Meljanac D., Mercati F., Pikutić D., Noncommutative spaces and Poincaré symmetry, Phys. Lett. B 766 (2017), 181-185, arXiv:1610.06716.
  31. Meljanac S., Meljanac D., Pachoł A., Pikutić D., Remarks on simple interpolation between Jordanian twists, J. Phys. A: Math. Theor. 50 (2017), 265201, 11 pages, arXiv:1612.07984.
  32. Meljanac S., Meljanac D., Samsarov A., Stojić M., $\kappa$-deformed Snyder spacetime, Modern Phys. Lett. A 25 (2010), 579-590, arXiv:0912.5087.
  33. Meljanac S., Pachoł A., Pikutić D., Twisted conformal algebra related to $\kappa$-Minkowski space, Phys. Rev. D 92 (2015), 105015, 8 pages, arXiv:1509.02115.
  34. Meljanac S., Stojić M., New realizations of Lie algebra kappa-deformed Euclidean space, Eur. Phys. J. C Part. Fields 47 (2006), 531-539, arXiv:hep-th/0605133.
  35. Ogievetsky O., Hopf structures on the Borel subalgebra of ${\rm sl}(2)$, Rend. Circ. Mat. Palermo (2) Suppl. (1994), 185-199.
  36. Pachoł A., Vitale P., $\kappa$-Minkowski star product in any dimension from symplectic realization, J. Phys. A: Math. Theor. 48 (2015), 445202, 16 pages, arXiv:1507.03523.
  37. Stolin A.A., Kulish P.P., New rational solutions of Yang-Baxter equation and deformed Yangians, Czechoslovak J. Phys. 47 (1997), 123-129, arXiv:q-alg/9608011.
  38. Tolstoy V.N., Quantum deformations of relativistic symmetries, arXiv:0704.0081.
  39. Tolstoy V.N., Twisted quantum deformations of Lorentz and Poincaré algebras, arXiv:0712.3962.

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