Capabilities of the Lie Tools Package

Existing software packages, such as LiE [21] or Maple's liesymm package [48,49], are very specialized and do not provide the functionality of LTP. Among other functions, LTP greatly automates and simplifies the following computations:

• Construction of ordered bases for free Lie algebras of indeterminates (Hall bases).
• Simplification of completely general Lie algebraic expressions with symbolic coefficients.
• Composition of exponential mappings involving indeterminates by means of Dynkin's expression for the Campbell-Baker-Hausdorff (CBH) formula.
• Construction of the Wei-Norman equations of logarithmic coordinates of flows on nilpotent Lie groups.

The above capabilities are unique to LTP and, to the best of our knowledge, are not provided by other software such as, for example, the liealg package, which was recently developed by Yuly Billig and Matthias Mazzag¹, to perform specific calculations involving Kac-Moody and Virasoro algebras, and their representations. None of the mature computer algebra systems (CAS), such as Axiom (former Scratchpad II by R. D. Jenks and D. Yun, IBM Watson Laboratories), Derive (D. R. Stoutemyer), Macsyma (Math Lab Group, MIT), Maple (B. Char, Waterloo Maple, Inc.), Mathematica (Wolfram Research, Inc.) or Reduce (A. C. Hearn), provide toolboxes with the functionality of LTP. For surveys and comparative reviews of the different CAS, the reader is referred to [6,46], the references in [38], and the information on symbolic computation available through Internet sites, such as the comprehensive Computer Algebra Information Network (CAIN)² or the Symbolic Mathematical Computation Information Center³.