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


SIGMA 8 (2012), 013, 15 pages      arXiv:1006.0478      http://dx.doi.org/10.3842/SIGMA.2012.013

Exponential Formulas and Lie Algebra Type Star Products

Stjepan Meljanac a, Zoran Škoda a and Dragutin Svrtan b
a) Division for Theoretical Physics, Institute Rudjer Bošković, Bijenička 54, P.O. Box 180, HR-10002 Zagreb, Croatia
b) Department of Mathematics, Faculty of Natural Sciences and Mathematics, University of Zagreb, HR-10000 Zagreb, Croatia

Received May 26, 2011, in final form March 01, 2012; Published online March 22, 2012

Abstract
Given formal differential operators $F_i$ on polynomial algebra in several variables $x_1,\ldots,x_n$, we discuss finding expressions $K_l$ determined by the equation $\exp(\sum_i x_i F_i)(\exp(\sum_j q_j x_j)) = \exp(\sum_l K_l x_l)$ and their applications. The expressions for $K_l$ are related to the coproducts for deformed momenta for the noncommutative space-times of Lie algebra type and also appear in the computations with a class of star products. We find combinatorial recursions and derive formal differential equations for finding $K_l$. We elaborate an example for a Lie algebra $su(2)$, related to a quantum gravity application from the literature.

Key words: star product; exponential expression; formal differential operator.

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References

  1. Amelino-Camelia G., Arzano M., Coproduct and star product in field theories on Lie-algebra noncommutative space-times, Phys. Rev. D 65 (2002), 084044, 8 pages, hep-th/0105120.
  2. Arnal D., Cortet J.C., $*$-products in the method of orbits for nilpotent groups, J. Geom. Phys. 2 (1985), 83-116.
  3. Arnal D., Cortet J.C., Molin P., Pinczon G., Covariance and geometrical invariance in $*$ quantization, J. Math. Phys. 24 (1983), 276-283.
  4. Aschieri P., Lizzi F., Vitale P., Twisting all the way: from classical mechanics to quantum fields, Phys. Rev. D 77 (2008), 025037, 16 pages, arXiv:0708.3002.
  5. Barron K., Huang Y.Z., Lepowsky J., Factorization of formal exponentials and uniformization, J. Algebra 228 (2000), 551-579, math.QA/9908151.
  6. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Physics 111 (1978), 61-110.
  7. Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D., Deformation theory and quantization. II. Physical applications, Ann. Physics 111 (1978), 111-151.
  8. Blasiak P., Flajolet P., Combinatorial models of creation-annihilation, Ann. Physics 65 (2011), Art. B65c, 78 pages, arXiv:1010.0354.
  9. Borowiec A., Pacho A., $\kappa$-Minkowski spacetimes and DSR algebras: fresh look and old problems, SIGMA 6 (2010), 086, 31 pages, arXiv:1005.4429.
  10. Dimitrijević M., Meyer F., Möller L., Wess J., Gauge theories on the $\kappa$-Minkowski spacetime, Eur. Phys. J. C Part. Fields 36 (2004), 117-126, hep-th/0310116.
  11. Durov N., Meljanac S., Samsarov A., Škoda Z., A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, J. Algebra 309 (2007), 318-359, math.RT/0604096.
  12. Freidel L., Livine E.R., 3D quantum gravity and effective noncommutative quantum field theory, Phys. Rev. Lett. 96 (2006), 221301, 4 pages, hep-th/0512113.
  13. Freidel L., Majid S., Noncommutative harmonic analysis, sampling theory and the Duflo map in 2+1 quantum gravity, Classical Quantum Gravity 25 (2008), 045006, 37 pages, hep-th/0512113.
  14. Halliday S., Szabo R.J., Noncommutative field theory on homogeneous gravitational waves, J. Phys. A: Math. Gen. 39 (2006), 5189-5225, hep-th/0602036.
  15. Kathotia V., Kontsevich's universal formula for deformation quantization and the Campbell-Baker-Hausdorff formula, Internat. J. Math. 11 (2000), 523-551, math.QA/9811174.
  16. Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), 157-216, q-alg/9709040.
  17. Meljanac S., Krešić-Jurić S., Stojić M., Covariant realizations of kappa-deformed space, Eur. Phys. J. C Part. Fields 51 (2007), 229-240, hep-th/0702215.
  18. Meljanac S., Škoda Z., Leibniz rules for enveloping algebras, arXiv:0711.0149, the latest version available at http://www.irb.hr/korisnici/zskoda/scopr5.pdf.
  19. Meljanac S., Stojić M., New realizations of Lie algebra kappa-deformed Euclidean space, Eur. Phys. J. C Part. Fields 47 (2006), 531-539, hep-th/0605133.
  20. Raševski P.K., Associative superenvelope of a Lie algebra and its regular representation and ideals, Trudy Moskov. Mat. Obšč. 15 (1966), 3-54.
  21. Škoda Z., Heisenberg double versus deformed derivatives, Internat. J. Modern Phys. A 26 (2011), 4845-4854, arXiv:0909.3769.
  22. Škoda Z., Twisted exterior derivatives for enveloping algebras, arXiv:0806.0978.

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