Symmetry Operators and Solutions to Differential Equations in Algebra

摘要

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.