Distortion of the Poisson Bracket by the Noncommutative Planck Constants

1. Alekseev A., Kosmann-Schwarzbach Y.: Manin pairs and moment maps. J. Diff. Geom. 56(1), 133–165 (2000)

   MathSciNet  MATH  Google Scholar 

2. Alekseev A., Kosmann-Schwarzbach Y., Meinrenken E.: Quasi-Poisson manifolds. Canad. J. Math. 54(1), 3–29 (2002)

   Article  MathSciNet  MATH  Google Scholar 

3. Beggs E. J., Majid S.: Semiclassical differential structures. Pacific J. Math. 224(1), 1–44 (2006)

   Article  MathSciNet  MATH  Google Scholar 

4. Van den Bergh M.: Double Poisson algebras. Trans. Amer. Math. Soc. 360(11), 5711–5769 (2008)

   Article  MathSciNet  MATH  Google Scholar 

5. Bocklandt R., Le Bruyn L.: Necklace Lie algebras and noncommutative symplectic geometry. Math. Z. 240(1), 141–167 (2002)

   Article  MathSciNet  MATH  Google Scholar 

6. Castro C.: On modified Weyl-Heisenberg algebras, noncommutativity, matrix-valued Planck constant and QM in Clifford spaces. J. Phys. A 39(45), 14205–14229 (2006)

   Article  MathSciNet  ADS  MATH  Google Scholar 

7. Chaichian M., Demichev A.P., Kulish P.P.: Quasi-classical limit in q-deformed systems, non-commutativity and the q-path integral. Phys. Lett. A 233(4–6), 251–260 (1997)

   Article  MathSciNet  ADS  MATH  Google Scholar 

8. Connes, A.: Cyclic cohomology and noncommutative differential geometry. Proceedings of the International Congress of Mathematicians, Vol. 1, (Berkeley, Calif., 1986), Providence, RI: Amer. Math. Soc., 1987, pp. 879–889

9. Crawley-Boevey W., Etingof P., Ginzburg V.: Noncommutative geometry and quiver algebras. Adv. Math. 209(1), 274–336 (2007)

   Article  MathSciNet  MATH  Google Scholar 

10. Drinfel’d, V.G.: Quasi-Hopf algebras. (Russian) Algebra i Analiz 1(6), 114–148 (1989), translation in Leningrad Math. J. 1(6), 1419–1457 (1990)

11. Farkas D.R.: A ring-theorist’s description of Fedosov quantization. Lett. Math. Phys. 51(3), 161–177 (2000)

    Article  MathSciNet  MATH  Google Scholar 

12. Fialowski A., Penkava M.: Deformation theory of infinity algebras. J. Algebra 255(1), 59–88 (2002)

    Article  MathSciNet  MATH  Google Scholar 

13. Ginzburg V.: Non-commutative symplectic geometry, quiver varieties, and operads. Math. Res. Lett. 8(3), 377–400 (2001)

    MathSciNet  MATH  Google Scholar 

14. Gurevich, D.I.: Algebraic aspects of the quantum Yang-Baxter equation. (Russian) Algebra i Analiz 2(4), 119–148 (1990), translation in Leningrad Math. J. 2(4), 801–828 (1991)

15. Gurevich, D.I., Pyatov, P.N., Saponov, P.A.: Representation theory of an algebra of a (modified) reflection equation of GL(m|n) type. (Russian) Algebra i Analiz 20(2), 70–133 (2008), translation in St. Petersburg Math. J. 20(2), 213–253 (2009)

16. Hartwig J.T., Larsson D., Silvestrov S.D.: Deformations of Lie algebras using σ-derivations. J. Algebra 295(2), 314–361 (2006)

    Article  MathSciNet  MATH  Google Scholar 

17. Hirshfeld A.C., Henselder P.: Star products and quantum groups in quantum mechanics and field theory. Ann. Physics 308(1), 311–328 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

18. Kapranov M.: Noncommutative geometry based on commutator expansions. J. Reine Angew. Math. 505, 73–118 (1998)

    Article  MathSciNet  MATH  Google Scholar 

19. Karasev, M.V., Maslov, V.P.: Nonlinear Poisson brackets. Geometry and quantization. Translated from the Russian by A. Sossinsky and M. Shishkova. Translations of Mathematical Monographs, 119. Providence, RI: Amer. Math. Soc., 1993

20. Kontsevich M.: Deformation quantization of Poisson manifolds, I. Lett. Math. Phys. 66, 157–216 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

21. Kontsevich, M.: Formal (non)commutative symplectic geometry. The Gel’fand Mathematical Seminars, 1990–1992, Boston, MA: Birkhuser Boston, 1993, pp. 173–187

22. Kubo F.: Finite-dimensional non-commutative Poisson algebras. J. Pure Appl. Algebra 113(3), 307–314 (1996)

    Article  MathSciNet  MATH  Google Scholar 

23. Le Bruyn, L.: Noncommutative geometry and Cayley-smooth orders. In: Pure and Applied Mathematics (Boca Raton), 290. Boca Raton, FL: Chapman & Hall/CRC, 2008

24. López Peña J., Panaite F., Van Oystaeyen F.: General twisting of algebras. Adv. Math. 212(1), 315–337 (2007)

    Article  MathSciNet  MATH  Google Scholar 

25. Maslov V.P.: An application of the method of ordered operators to the problem of obtaining exact solutions. (Russian) Teoret. Mat. Fiz. 33(2), 185–209 (1977)

    MATH  Google Scholar 

26. Maslov, V.P.: Operational methods. Translated from the Russian by V. Golo, N. Kulman, G. Voropaeva. Moscow: Mir Publishers, 1976

27. Maslov, V.P.: Nonstandard characteristics in asymptotic problems. (Russian) Usp. Mat. Nauk 38(6) (234), 3–36 (1983)

28. Newmann M.H.A.: On theories with a combinatorial definition of “equivalence”. Ann. of Math. 43, 223–243 (1942)

    Article  MathSciNet  Google Scholar 

29. Van Oystaeyen, F., Verschoren, A.: Noncommutative algebraic geometry. An introduction. Lecture Notes in Mathematics, 887. Berlin: Springer-Verlag, 1981

30. Panaite F., Van Oystaeyen F.: Quasi-Hopf algebras and representations of octonions and other quasialgebras. J. Math. Phys. 45(10), 3912–3929 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

31. Penkava, M.: L-infinity algebras and their cohomology. http://arxiv.org/abs/q-alg/9512014v1 , 1995

32. Pichereau A., Van de Weyer G.: Double Poisson cohomology of path algebras of quivers. J. Algebra 319(5), 2166–2208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

33. Richard L., Silvestrov S.D.: Quasi-Lie structure of σ-derivations of \({\mathbb{C}[t^{\pm 1}]}\) . J. Alg. 319(3), 1285–1304 (2008)

    Article  MathSciNet  MATH  Google Scholar 

34. Xu P.: Noncommutative Poisson algebras. Amer. J. Math. 116(1), 101–125 (1994)

    Article  MathSciNet  MATH  Google Scholar 

35. Yao Y., Ye Y., Zhang P.: Quiver Poisson algebras. J. Algebra 312(2), 570–589 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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