Computing the Baker-Campbell-Hausdorff series and the Zassenhaus product

The Baker-Campbell-Hausdorff (BCH) series and the Zassenhaus product are of fundamental importance for the theory of Lie groups and their applications in physics. In this paper, various methods for the computation of the terms in these expansions are compared, and a new efficient approach for the calculation of the Zassenhaus terms is introduced. Mathematica implementations for the most efficient algorithms are provided together with comparisons of execution times. Furthermore, we study two maps which translate the polynomial representation of the BCH and Zassen-haus terms into representations in terms of nested commutators. The first of these maps yields the well known Dynkin-Specht-Wever representation of the BCH terms while the second one generates a commuta-tor representation which involves fewer terms than the Dynkin-Specht-Wever representation.