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Grassmann variables on quantum spaces

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Abstract.

Attention is focused on antisymmetrized versions of quantum spaces that are of particular importance in physics, i.e. two-dimensional quantum plane, q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. For each case standard techniques for dealing with q-deformed Grassmann variables are developed. Formulae for multiplying supernumbers are given. The actions of symmetry generators and fermionic derivatives upon antisymmetrized quantum spaces are calculated. The complete Hopf structure for all types of quantum space generators is written down. From the formulae for the coproduct a realization of the L-matrices in terms of symmetry generators can be read off. The L-matrices together with the action of symmetry generators determine how quantum spaces of different type have to be fused together.

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References

  1. W. Heisenberg, Ann. Phys. 32, 20 (1938)

    Google Scholar 

  2. H.S. Snyder, Phys. Rev. 71, 38 (1947)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. P.P. Kulish, N.Y. Reshetikhin, J. Sov. Math. 23, 2345 (1983)

    Article  Google Scholar 

  4. V.G. Drinfeld, Quantum groups, in Proceedings of the International Congress of Mathematicians, 1986, edited by A.M. Gleason (Amer. Math. Soc., 1986), p. 798

  5. V.G. Drinfeld, Sov. Math. Dokl. 32, 254 (1985)

    Google Scholar 

  6. M. Jimbo, Lett. Math. Phys. 10, 63 (1985)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. S.L. Woronowicz, Commun. Math. Phys. 111, 613 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Y.J. Manin, Quantum groups and Non-Commutative Geometry(Centre de Recherche Mathematiques, Montreal 1988)

  9. N.Yu. Reshetikhin, L.A. Takhtadzhyan, L.D. Faddeev, Leningrad Math. J. 1, 193 (1990)

    MATH  MathSciNet  Google Scholar 

  10. U. Carow-Watamura, M. Schlieker, M. Scholl, S. Watamura, Z. Phys. C 48, 159 (1990)

    Article  MathSciNet  Google Scholar 

  11. P. Podles, S.L. Woronowicz, Commun. Math. Phys. 130, 381 (1990)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. W.B. Schmidke, J. Wess, B. Zumino, Z. Phys. C 52, 471 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. S. Majid, J. Math. Phys. 32, 3246 (1991)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. A. Lorek, W. Weich, J. Wess, Z. Phys. C 76, 375 (1997) [q-alg/9702025]

    Article  MathSciNet  Google Scholar 

  15. M. Fichtmüller, A. Lorek, J. Wess, Z. Phys. C 71, 533 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  16. B.L. Cerchiai, J. Wess, Eur. Phys. J. C 5, 553 (1998) [math.QA/9801104]

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. S. Majid, Int. J. Mod. Phys. A 5, 4689 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. H. Grosse, C. Klimčik, P. Prešnajder, Int. J. Theor. Phys. 35, 231 (1996) [hep-th/9505175]

    Article  MATH  MathSciNet  Google Scholar 

  19. R. Oeckl, Commun. Math. Phys. 217, 451 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  20. C. Blohmann, Eur. Phys. J. C 30, 435 (2003) [hep-th/0111172]

    Article  ADS  MATH  MathSciNet  Google Scholar 

  21. H. Wachter, M. Wohlgenannt, Eur. Phys. J. C 23, 761 (2002) [hep-th/0103120]

    Article  ADS  MathSciNet  Google Scholar 

  22. C. Bauer, H. Wachter, Eur. Phys. J. C 31, 261 (2003) [math-ph/0201023]

    Article  ADS  MATH  Google Scholar 

  23. H. Wachter, Eur. Phys. J. C 32, 281 (2004) [hep-th/0206083]

    Article  ADS  MathSciNet  Google Scholar 

  24. H. Wachter, Eur. Phys. J. C 37, 379 (2004) [hep-th/0401113]

    Article  ADS  MathSciNet  Google Scholar 

  25. H. Wachter, q-Translations on quantum spaces [hep-th/0410205]

  26. H. Wachter, Braided products for quantum spaces [math-ph/0509018]

  27. S. Majid, Foundations of quantum group theory (University Press, Cambridge 1995)

  28. M. Chaichian, A.P. Demichev, Introduction to quantum groups (World Scientific, Singapore 1996)

  29. S. Majid, Suppl. Rend. Circ. Mat. Palermo, Ser. II, 26, 197 (1991)

  30. S. Majid, Lec. Notes Pure Appl. Math. 158, 55 (1994)

    MATH  MathSciNet  Google Scholar 

  31. S. Mac Lane, Categories for the working mathematician (Springer, 1974)

  32. A. Klimyk, K. Schmüdgen, Quantum groups and their representations (Springer, Berlin 1997)

  33. A. Lorek, W.B. Schmidke, J. Wess, Lett. Math. Phys. 31, 279 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  34. J. Wess, q-deformed Heisenberg Algebras, in Proceedings of the 38. Internationale Universitätswochen für Kern- und Teilchenphysik, Lect. Notes in Phys. no. 543, Schladming, 2000, edited by H. Gausterer, H. Grosse, L. Pittner (Springer, 2000) [math-phy/9910013]

  35. J. Wess, B. Zumino, Nucl. Phys. B Suppl. 18, 302 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  36. U. Carow-Watamura, M. Schlieker, S. Watamura, Z. Phys. C 49, 439 (1991)

    Article  MathSciNet  Google Scholar 

  37. X.C. Song, Z. Phys. C 55, 417 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  38. T. Tanisaki, Killing Forms, Harish-Chandra homomorphisms and universal R-matrices for quantum algebras, in Infinite Algebras, edited by A. Tsuchiya, T. Eguchi, M. Jimbo (World Scientific, Singapore 1992)

  39. S. Majid, J. Math. Phys. 34, 2054 (1993) [hep-th/9210141]

    ADS  Google Scholar 

  40. S. Majid, J. Math. Phys. 36, 4436 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  41. S. Majid, J. Geom. Phys. 22, 14 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  42. M. Schlieker, W. Scholl, Z. Phys. C 52, 471 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  43. H. Ocampo, Z. Phys. C 70, 525 (1996)

    Article  Google Scholar 

  44. O. Ogievetsky, W.B. Schmidke, J. Wess, B. Zumino, Commun. Math. Phys. 150, 495 (1992)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  45. J. Lukierski, A. Nowicki, H. Ruegg, Phys. Lett. B 293, 344 (1992)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  46. L. Castellani, Differential Calculus onISO q (N), Quantum Poincaré Algebra and q-Gravity [hep-th/9312179]

  47. V.K. Dobrev, Phys. Lett. B 341, 133 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  48. M. Chaichian, A.P. Demichev, J. Math. Phys. 36, 398 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  49. M. Chaichian, P.P. Kulish, K. Nishijima, A. Tureanu, Phys. Lett. B 604, 98 (2004) [hep-th/0408062]

    Article  ADS  MathSciNet  Google Scholar 

  50. F. Koch, E. Tsouchnika, Nucl. Phys. B 717, 387 (2005) [hep-th/0409012]

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Sektion Physik, Ludwig-Maximilians-Universität, Theresienstr. 37, 80333, München, Germany

    D. Mikulovic, A. Schmidt & H. Wachter

Authors
  1. D. Mikulovic
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  2. A. Schmidt
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  3. H. Wachter
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Correspondence to D. Mikulovic.

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Arrival of the final proofs: 6 December 2005

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Mikulovic, D., Schmidt, A. & Wachter, H. Grassmann variables on quantum spaces. Eur. Phys. J. C 45, 529–544 (2006). https://doi.org/10.1140/epjc/s2005-02441-4

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  • DOI: https://doi.org/10.1140/epjc/s2005-02441-4

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