Using LTP: Some Practical Examples

The Maple code for the examples presented in this section is distributed with LTP, which may be obtained at: http://www.cim.mcgill.ca/$\sim$migueltt/ltp/ltp.html. Any Lie product which is written in terms of the algebra generators only, will henceforth be referred to as a Lie monomial. By the property of distributivity over scalar multiplication, an arbitrary Lie bracket is a product of a symbolic coefficient and a Lie monomial. The main and auxiliary functions provided by LTP are summarized in Table 1, p. , and Table 2, p. , respectively. Auxiliary functions are invoked by the main functions, but are also made directly available to the user to allow for perusal of intermediate results. Such an organization of the package facilitates the addition of new functions. See Section 6, p. , for details on the function's syntax, their algorithmic implementation, and other aspects. Prior to invoking any function in the package, two special variables need to be declared, under arbitrary names, to signify: the number of generators in the Lie algebra $L(\bar{X}_m)$ and its assumed order of nilpotency. The values of these variables are limited only by the available computer memory. The examples presented in the next two sections consider a set of Lie algebra generators $\bar{X}_3=(X_1,X_2,X_3)$ and a HB, denoted by $B$, for a nilpotent Lie algebra $L_4(\bar{X}_3)$ with degree of nilpotency $k=4$. The generators $\bar{X}_3$ and the basis $B$ are easily obtained by executing the package function phb(3,4); the resulting basis $B$ is shown in § 5.2.