Introduction

The Lie Tools Package (LTP) is a collection of routines whose purpose is to make easier the task of manipulating Lie algebraic expressions arising in a variety of problems in engineering and mathematical physics, such as the solution of right-invariant differential equations evolving on Lie groups. Lie theory is a powerful tool, helpful in the analysis and design of modern nonlinear control laws for dynamical systems, nonlinear filters, and the study of particle dynamics. The practical application of Lie theory often results in highly complex symbolic expressions that are difficult to handle efficiently without the aid of a computer software tool. The aim of this package is also to facilitate and encourage further research relying on Lie algebraic computations. LTP is implemented in Maple$^{\circledR}$ and can be employed for computations involving Lie algebras of arbitrary type as it is constructed using a free Lie algebra of indeterminates as its base. The results obtained with the help of LTP can subsequently be projected onto the specific Lie algebra arising in the concrete application of interest by the use of an adequately constructed Lie algebra homomorphism. This document is not intended to teach the user the basic concepts in Lie algebra and Lie group theory. Basic references to Lie theory can be found in Section 1.4. A brief review of the basic notions and LTP formalism is presented in Section 3.