Automatic determination of optimal systems of Lie subalgebras: The package SymboLie

Lie groups of point symmetries of partial differential equations constitute a fundamental tool for constructing group–invariant solutions. The number of subgroups is potentially infinite and so the number of group–invariant solutions. An important goal is a classification in order to have an optimal system of inequivalent group–invariant solutions from which all other solutions can be derived by action of the group itself. In turn, a classification of inequivalent subgroups induces a classification of inequivalent Lie subalgebras, and vice versa. A general method for classifying the Lie subalgebras of a finite–dimensional Lie algebra relies on the use of inner automorphisms. We present a novel effective algorithm that can automatically determine optimal systems of Lie subalgebras of a generic finite–dimensional Lie algebra; here, we limit the analysis to one–dimensional Lie subalgebras, though the same approach still works well for higher dimensional Lie subalgebras. The algorithm is implemented in the computer algebra system Wolfram Mathematica (TM) and illustrated by means of some examples.