An Evolutionary Approach to the Automatic Classification of Automorphisms of Lower-Dimensional Lie Algebras

We implement two algorithms in MATHEMATICA for classifying automorphisms of lower-dimensional non-Abelian Lie algebras. The first algorithm is a brute-force approach whereas the second is an evolutionary strategy. These algorithms are delivered as the MATHEMATICA package cwsAutoClass. In order to facilitate the application of this package to symmetry Lie algebras of differential equations, we also provide a package, cwsLieSymTools, for manipulating finite-dimensional Lie algebras of vector fields. In particular, this package allows the computations of Lie brackets, structure constants, and the visualization of commutator tables. Several examples are provided to illustrate the pertinence of our approach. These computational experiments show that the evolutionary strategy outperformed consistently the brute-force method.