Physics of Particles and Nuclei

, Volume 43, Issue 5, pp 663–665 | Cite as

On the variational noncommutative poisson geometry

  • A. V. Kiselev


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.


Jacobi Identity Leibniz Rule Evolutionary Vector Schouten Bracket Cyclic Invariance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Kontsevich, “Formal (Non)commutative Symplectic Geometry,” in The Gel’fand Math. Sem., 1990–1992 (Birkhäuser, Boston, 1993), pp. 173–187.CrossRefGoogle Scholar
  2. 2.
    P. J. Olver and V. V. Sokolov, “Integrable Evolution Equations on Associative Algebras,” Commun. Math. Phys. 193, 245–268 (1998).MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 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. 4.
    I. Krasil’shchik and A. Verbovetsky, “Geometry of Jet Spaces and Integrable Systems,” J. Geom. Phys. 61, 1633–1674 (2011).MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 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.Google Scholar
  6. 6.
    M. Kontsevich, “Deformation Quantization of Poisson Manifolds. I,” Lett. Math. Phys. 66, 157–216 (2003). arXiv:q-alg/9709040MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 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).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  • A. V. Kiselev

There are no affiliations available

Personalised recommendations