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On the variational noncommutative poisson geometry

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Abstract

We outline the notions and concepts of the calculus of variational multivectors within the Poisson formalism over the spaces of infinite jets of mappings from commutative (non)graded smooth manifolds to the factors of noncommutative associative algebras over the equivalence under cyclic permutations of the letters in the associative words. We state the basic properties of the variational Schouten bracket and derive an interesting criterion for (non)commutative differential operators to be Hamiltonian (and thus determine the (non)commutative Poisson structures). We place the noncommutative jet-bundle construction at hand in the context of the quantum string theory.

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References

  1. M. Kontsevich, “Formal (Non)commutative Symplectic Geometry,” in The Gel’fand Math. Sem., 1990–1992 (Birkhäuser, Boston, 1993), pp. 173–187.

    Chapter  Google Scholar 

  2. P. J. Olver and V. V. Sokolov, “Integrable Evolution Equations on Associative Algebras,” Commun. Math. Phys. 193, 245–268 (1998).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. P. J. Olver, Applications of Lie Groups to Differential Equations, in Graduate Texts in Math, 2nd ed. (Springer, New York, 1993), Vol. 107.

    Google Scholar 

  4. I. Krasil’shchik and A. Verbovetsky, “Geometry of Jet Spaces and Integrable Systems,” J. Geom. Phys. 61, 1633–1674 (2011).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. A. V. Kiselev, “Homological Evolutionary Vector Fields in Korteweg — De Vries, Liouville, Maxwell, and Several Other Models,” J. Phys.: Conf. Ser., vol. 343, no. 012058, pp. 1–20.

  6. M. Kontsevich, “Deformation Quantization of Poisson Manifolds. I,” Lett. Math. Phys. 66, 157–216 (2003). arXiv:q-alg/9709040

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. A. V. Kiselev and J. W. van de Leur, “Variational Lie Algebroids and Homological Evolutionary Vector Fields,” Theor. Math. Phys. 167, 772–784 (2011).

    Article  Google Scholar 

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Authors
  1. A. V. Kiselev
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Kiselev, A.V. On the variational noncommutative poisson geometry. Phys. Part. Nuclei 43, 663–665 (2012). https://doi.org/10.1134/S1063779612050188

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