Abstract
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 Zn and consider the algebras D*(A) and ∧*(A) within the tensor category of Zn+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.
Similar content being viewed by others
References
R. Bott, Canadian Math. Bulletin, 28 (2), 129–164 (1985).
V. Chari and A. Pressley, A Guide to Quantum Groups (Cambridge University Press, 1994).
H. L. Huru, Lobachevskii J. of Mathematics 24, 13–42 (2006).
H. L. Huru, Lobachevskii J. of Mathematics 25, 131–185 (2007).
H. L. Huru, Russian Mathematics (Iz. VUZ) 52 (4), 65–76 (2008).
H. L. Huru, “Differential Equations—Geometry, Symmetries and Integrability,” B. Kruglikov, V. Lychagin, and E. Straume, Abel Symposia 5, 151–158 (2009).
J. Krasil’shchik, The Diffiety Institute Preprint Series, Preprint DIPS-1, 45 p. (1999).
I. S. Krasil’shchik and P. H. M. Kersten, Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations (Kluwer Academic Publisher, 2000).
V. V. Lychagin, C. R. Acad. Sci. Paris 318, 857–862 (1994).
V. V. Lychagin, Pergamon Nonlinear Analysis 47, 2621–2632 (2001).
Saunders Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (Springer, 1998).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huru, H. Braidings and quantizations of Maxwell’s equations. Lobachevskii J Math 36, 250–259 (2015). https://doi.org/10.1134/S1995080215030051
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080215030051