Lobachevskii Journal of Mathematics

, Volume 36, Issue 3, pp 250–259 | Cite as

Braidings and quantizations of Maxwell’s equations



This paper extends the grading and the symmetry of the Z-graded superalgebras of differential operators D*(A) and differential forms ∧*(A) on an associative algebra A. We consider the associative algebra A = C[x 1,..., x n ] as graded by Z n and consider the algebras D*(A) and ∧*(A) within the tensor category of Z n+1-graded modules. Any braiding/symmetry and any categorical quantization are completely described in terms of this grading.

We explore the quantizations of these algebras with corresponding quantized multiplications. We further describe the braided and quantized braided structures of the algebras’ derivations, curvatures, differential forms, inner derivations and exterior derivations.

We apply these results to the associative algebra A = C[x, y, z, t] and investigate and compare the quantizations of the electromagnetic 2-form and the Maxwell’s equations expressed in terms of differential forms in two settings: in the tensor categories of Z4-graded and Z5-graded modules.

Keywords and phrases

Braidings Categorical Quantizations Polynomial Algebra Differential Algebras Electromagnetic 2-form Maxwells equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Bott, Canadian Math. Bulletin, 28 (2), 129–164 (1985).MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    V. Chari and A. Pressley, A Guide to Quantum Groups (Cambridge University Press, 1994).MATHGoogle Scholar
  3. 3.
    H. L. Huru, Lobachevskii J. of Mathematics 24, 13–42 (2006).MATHMathSciNetGoogle Scholar
  4. 4.
    H. L. Huru, Lobachevskii J. of Mathematics 25, 131–185 (2007).MATHMathSciNetGoogle Scholar
  5. 5.
    H. L. Huru, Russian Mathematics (Iz. VUZ) 52 (4), 65–76 (2008).MATHMathSciNetGoogle Scholar
  6. 6.
    H. L. Huru, “Differential Equations—Geometry, Symmetries and Integrability,” B. Kruglikov, V. Lychagin, and E. Straume, Abel Symposia 5, 151–158 (2009).MathSciNetGoogle Scholar
  7. 7.
    J. Krasil’shchik, The Diffiety Institute Preprint Series, Preprint DIPS-1, 45 p. (1999).Google Scholar
  8. 8.
    I. S. Krasil’shchik and P. H. M. Kersten, Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations (Kluwer Academic Publisher, 2000).MATHCrossRefGoogle Scholar
  9. 9.
    V. V. Lychagin, C. R. Acad. Sci. Paris 318, 857–862 (1994).MATHMathSciNetGoogle Scholar
  10. 10.
    V. V. Lychagin, Pergamon Nonlinear Analysis 47, 2621–2632 (2001).MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (Springer, 1998).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.University of Tromsø–The Arctic University of NorwayTromsøNorway

Personalised recommendations