represent a Lie Algebra of symmetry generators in terms of derived algebras
list of symmetry generators
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 Gi of G are defined recursively as follows:
1 is G;
G is the Lie Algebra obtained by taking all possible commutators of 1;
in general, Gi+1 is the Lie Algebra obtained by taking all possible commutators of Gi.
Since G is assumed to be finite, there exists a positive integer n with the following properties:
(i) Gn+1 = Gn
(ii) n is the smallest integer possessing property (i).
solve_group returns a list L of n+1 lists of symmetries with the following properties:
The symmetries inside the list L1 form the basis for Gn
The symmetries inside the lists L1 and L2 together form the basis for Gn−1.
In general, the symmetries inside the first n+1−i lists of L together form the basis for Gi.
In other words, map(op, L[1..n+1-i]) is a basis for Gi.
The group G is solvable if Gn is the zero group. If G is solvable then the first element of the returned list L 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](..).
G ≔ ξ⁡x,y,η⁡x,y
G20 ≔ 0,1,1,0
G21 ≔ 0,1,0,y
G ≔ 1,0,0,1,ⅇy,0
SL2 ≔ 0,1,0,y,0,y2
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