Background Information and Reference Material

The theory of Lie algebras and groups was originally conceived by the Norwegian mathematician Sophus Lie (1842-1899) as a tool for the solution of differential equations and has since then become a discipline in its own right. Lie theory brings together the mathematical disciplines of algebra and geometry to produce results relying on group-theoretic and differential geometric developments. Important basic references in Lie algebras and group theory are the book by V.S. Varadarajan and J.-P. Serre [43,36]. A basic reference which is intended to serve an audience of physicists and engineers is the book by R. Gilmore [13]. For a comprehensive review of the applications of the Lie theory also see [13] and the book by J. G. F. Belinfante [3], which also presents a surver of some computational methods. Results in Lie theory have proved essential in the study of kinematical symmetries in both classical and quantum mechanics [9,33], the construction of nonlinear filters [7,23], the analysis of dynamical systems, and the design of feedback control laws for nonlinear systems [32,16,27]. The use of Lie theory in the study of the symmetries of differential equations is described in [38] from a practical perspective. The application of Lie theory to the analysis and control of robotic systems is found in [27,35,32] and references therein. Despite the attention that Lie theory has received in a variety of fields, it has been limited mainly because of the complexity of the symbolic calculations, which are often prohibitively difficult to perform by hand, and the lack of adequate software capable of handling completely general symbolic Lie algebraic expressions.