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reduceLBT

 


  
Purpose 		 
Substitute Lie Bracket Table relations in a Lie polynomial written in terms of PHB elements.


Syntax e:=reduceLBT(x,B,lbt);

Description
This procedure is similar to reduceLB, it simplifies a Lie polynomial given in $x$ to a expression in terms of elements in the PHB in $B$, but additionally substitutes the dependent brackets in the PHB according to Lie bracket table relations passed to the procedure in $lbt$. The table defining the dependent brackets in terms of independent ones is passed as a list. In the case that all elements in the PHB are linearly independent, then $lbt$ should be an empty set or list, and the result returned by reduceLBT will be equal to that obtained by using reduceLB if there are no brackets of order higher than the degree of nilpotency. To make this clearer we stress the following characteristic of this procedure:
Unlike reduceLB, this procedure eliminates from the expression $x$ all those Lie brackets of order higher than the degree of nilpotency, i.e. sets them to zero.


Arguments $x$ $\textstyle \parbox{0.64\textwidth}{Any Lie polynomial.}$
$B$ The Philip Hall basis.
$lbt$ $\textstyle \parbox{0.64\textwidth}{The Lie bracket table (lbt) is a set 
 or li...
...ie bracket table could also be an empty list or set,
 e.g. \texttt{lbt:=\{\}}.}$

Examples
Consider the Lie polynomial x given by:
                  2
   x := 1/2 dg3 g1  (f0 &* (f0 &* f2))

         + dg4 g2 (f1 &* (f0 &* f1))
and the Lie bracket table sr,
> sr:={B[4]=B[1]+6*B[2]};

                   sr := {f0 &* f1 = f0 + 6 f1}
The function reduceLBT with the P. Hall basis given in the example for the phb function on page [*], yields
> reduceLBT(x,B,sr);

            2
 1/2 dg3 g1  (f0 &* (f0 &* f2)) - dg4 g2 f0 - 6 dg4 g2 f1


Implementation Notes
This function basically relies only on reduceLB and Maple's subs function. The algorithm is rather simple, the procedure repeatedly tries to replace the dependent relations in $x$ using the substitution relations in $lbt$ until there are no more dependent terms (brackets) in $x$. At every substitution iteration, the function reduceLB is called to simplify the expression.

next up previous contents
Next: ad Up: Function Reference Previous: codeCBHcf   Contents
Miguel Attilio Torres-Torriti 2004-05-31