Symmetry Operators and Solutions to Differential Equations in Algebra

  • Roei Nisim AsrafEmail author


The symmetry operator method for finding an analytical form for the solutions to a given PDE system is presented. The main tools for accomplishing this aim include computation and grading of the Lie algebra infinitesimals of symmetry groups admitted by a given PDE system, establishing fusion rules, and then checking the commutator relations after performing the grading. We discuss several examples of the differential systems of mathematical physics based on the splitting of heat and the Euler–Tricomi, generalized Cauchy–Riemann and Dirac equations in non-associative algebras.


Symmetry algebra Algebra graduating Polynomial solutions 

Mathematics Subject Classification

58J70 76M60 



I thank Prof. Y. Krasnov for valuable suggestions, useful advice, and for encouraging me to write this paper, to Prof. J. Schiff for helpful comments and for suggesting Theorem 5.3, and Mrs. M. Beller who edited this paper.


  1. 1.
    Colladay, D., Mcdonald, P., Mullins, D.: Quaternionic formulation of the Dirac equation. CPT and Lorentz Symmetry, pp. 199–203 (2010)Google Scholar
  2. 2.
    Dirac, P.A.M.: The quantum theory of the electron. Proc. R. Soc. A 117, 610–624 (1928)CrossRefGoogle Scholar
  3. 3.
    Eidelman, S.D., Krasnov, Y.: Operator theory, systems theory and scattering theory: multidimensional generalizations. Oper. Theory Adv. Appl., vol. 157, pp. 107–137. Birkhuser, Basel (2006)Google Scholar
  4. 4.
    Krasnov, Y.: Properties of ODEs and PDEs in algebras. Complex Anal. Oper. Theory 7(3), 623–634 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Miller, W.: Symmetry and Separation of Variables: Encyclopedia of Mathematics and Its Applications, vol. 4. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam (1977)Google Scholar
  6. 6.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, vol. 107. Springer, New York (1993)CrossRefGoogle Scholar
  7. 7.
    Olver, P.J.: Symmetry and explicit solutions of partial differential equations. Appl. Numer. Math. 10, 307–324 (1992)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Pierce, R.S.: Modules Over Commutative Regular Rings. Memories of the American Mathematical Society, No. 70. American Mathematical Society, Providence (1967)Google Scholar
  9. 9.
    Rowen, L.H.: Graduate Algebra: Noncommutative View. Graduate Studies in Mathematics, vol. 91. American Mathematical Society, Providence (2008)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael

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