J Supercomput (2012) 59:147–155 DOI 10.1007/s11227-010-0430-2 A computational study of a family of nilpotent Lie algebras Juan Núñez · Ángel F. Tenorio Published online: 14 April 2010 © Springer Science+Business Media, LLC 2010 Abstract This paper describes an algorithm to compute the law of the Lie algebra g n associated with the Lie group G n , formed of all the n × n upper-unitriangular matri- ces. The goal of this paper is to show the algorithm that computes the law of g n and its implementation using the symbolic computation package MAPLE. In addition, the complexity of the algorithm is described. Keywords Nilpotent Lie algebra · MAPLE · Algorithm · Complexity 1 Introduction At present, the relationship between Lie groups and Lie algebras has been exten- sively studied. A unique complex finite-dimensional Lie algebra is associated with a given complex finite-dimensional Lie group. This algebra can be constructed by considering the C-vector space of all left-invariant differentiable vector fields with its associated commutator (see [9, Sect. 2.3]). Conversely, given a finite-dimensional complex Lie algebra, it is possible to obtain a corresponding complex Lie group (see [9, Theorem 3.17.8]). Nevertheless, this fact cannot be generalized to the infinite- dimensional case because there exist infinite-dimensional Lie algebras that are not associated with any Lie groups (see [8]). Coming back to the finite-dimensional case, the existence of a Lie group associ- ated with a given Lie algebra does not imply the uniqueness of such a Lie group. This J. Núñez Dpto. Geometría y Topología, Facultad de Matemáticas, Universidad de Sevilla, Sevilla, Spain e-mail: jnvaldes@us.es Á.F. Tenorio ( ) Dpto. Economía, Métodos Cuantitativos e Historia Económica, Universidad Pablo de Olavide, Sevilla, Spain e-mail: aftenorio@upo.es