DEtools
solve_group
represent a Lie Algebra of symmetry generators in terms of derived algebras
Calling Sequence
Parameters
Description
Examples
solve_group(G, y(x))
G
-
list of symmetry generators
y(x)
dependent and independent variables
solve_group receives a list G of infinitesimals corresponding to symmetry generators that generate a finite dimensional Lie Algebra G, and returns a representation of the derived algebras of G.
Derived algebras of G are defined recursively as follows:
is G;
is the Lie Algebra obtained by taking all possible commutators of ;
in general, is the Lie Algebra obtained by taking all possible commutators of .
Since G is assumed to be finite, there exists a positive integer with the following properties:
(i) =
(ii) is the smallest integer possessing property (i).
solve_group returns a list of lists of symmetries with the following properties:
The symmetries inside the list form the basis for
The symmetries inside the lists and together form the basis for .
In general, the symmetries inside the first lists of together form the basis for .
In other words, map(op, L[1..n+1-i]) is a basis for .
The group G is solvable if is the zero group. If G is solvable then the first element of the returned list will be the empty list [].
This function is part of the DEtools package, and so it can be used in the form solve_group(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[solve_group](..).
See Also
canoni
DEtools/reduce_order
dsolve,Lie
equinv
eta_k
PDEtools
symgen
Xcommutator
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