Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs

We present an algorithm COMMUTATION.RELATIONS, which can calculate the commutation relations for the Lie symmetry algebra of symmetry operators for any system of PDEs. Unlike existing methods, COMMUTATION_RELATIONS does not depend on the heuristic process of integrating the associated differential equations for the symmetry operators (i.e. integrating the ‘determining equations’), An algorithm INITIAL-DATA, developed in previous work, is used to calculate lists of initial data which are in l-to-l correspondence with solutions of determining equations. COMMUTATION.RELATIONS exploits this correspondence by calculating commutators in terms of initial data. The method has been implemented in the symbolic language MAPLE and can be applied to both finiteand infinite-dimensional Lie symmetry algebras. We show how knowledge of the Lie symmetry algebra calculated by CoMMUTATION.RELATIONS can simplify the task of explicitly integrating determining equations.