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 $G^i$ 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, $G^{i+1}$ is the Lie Algebra obtained by taking all possible commutators of $G^i$.

Since G is assumed to be finite, there exists a positive integer $n$ with the following properties:

(i) $G^{n+1}$ = $G^n$

(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 $L_1$ form the basis for $G^n$

The symmetries inside the lists $L_1$ and $L_2$ together form the basis for $G^{n-1}$.

In general, the symmetries inside the first $n+1-i$ lists of $L$ together form the basis for $G^i$.

In other words, map(op, L[1..n+1-i]) is a basis for $G^i$.

The group G is solvable if $G^n$ 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](..).

$\mathrm{with}\left(\mathrm{DEtools}\right)\:$

$G≔\left[\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right]$

$G≔\left[\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right]$

$\mathrm{solve\_group}\left(G\,y\left(x\right)\right)$

$\left[\left[\right]\,\left[\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right]\right]$

$\mathrm{G20}≔\left[\left[0\,1\right]\,\left[1\,0\right]\right]$

$\mathrm{G20}≔\left[\left[0\,1\right]\,\left[1\,0\right]\right]$

$\mathrm{Xcommutator}\left(\mathrm{op}\left(\mathrm{G20}\right)\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=0\right]$

$\mathrm{solve\_group}\left(\mathrm{G20}\,y\left(x\right)\right)$

$\left[\left[\right]\,\left[\left[0\,1\right]\,\left[1\,0\right]\right]\right]$

$\mathrm{G21}≔\left[\left[0\,1\right]\,\left[0\,y\right]\right]$

$\mathrm{G21}≔\left[\left[0\,1\right]\,\left[0\,y\right]\right]$

$\mathrm{Xcommutator}\left(\mathrm{op}\left(\mathrm{G21}\right)\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=1\right]$

$\mathrm{solve\_group}\left(\mathrm{G21}\,y\left(x\right)\right)$

$\left[\left[\right]\,\left[\left[0\,1\right]\right]\,\left[\left[0\,y\right]\right]\right]$

$G≔\left[\left[1\,0\right]\,\left[0\,1\right]\,\left[\exp \left(y\right)\,0\right]\right]$

$G≔\left[\left[1\,0\right]\,\left[0\,1\right]\,\left[{\ⅇ}^y\,0\right]\right]$

$\mathrm{Xcommutator}\left(G\left[1\right]\,G\left[2\right]\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=0\right]$

$\mathrm{Xcommutator}\left(G\left[1\right]\,G\left[3\right]\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=0\right]$

$\mathrm{Xcommutator}\left(G\left[2\right]\,G\left[3\right]\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}={\ⅇ}^y\,\mathrm{\_\η}=0\right]$

$\mathrm{solve\_group}\left(G\,y\left(x\right)\right)$

$\left[\left[\right]\,\left[\left[{\ⅇ}^y\,0\right]\right]\,\left[\left[1\,0\right]\,\left[0\,1\right]\right]\right]$

$\mathrm{SL2}≔\left[\left[0\,1\right]\,\left[0\,y\right]\,\left[0\,y^2\right]\right]$

$\mathrm{SL2}≔\left[\left[0\,1\right]\,\left[0\,y\right]\,\left[0\,y^2\right]\right]$

$\mathrm{Xcommutator}\left(\mathrm{SL2}\left[1\right]\,\mathrm{SL2}\left[2\right]\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=1\right]$

$\mathrm{Xcommutator}\left(\mathrm{SL2}\left[1\right]\,\mathrm{SL2}\left[3\right]\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=2y\right]$

$\mathrm{Xcommutator}\left(\mathrm{SL2}\left[2\right]\,\mathrm{SL2}\left[3\right]\,y\left(x\right)\right)$

$\left[\mathrm{\_\ξ}=0\,\mathrm{\_\η}=y^2\right]$

$\mathrm{solve\_group}\left(\mathrm{SL2}\,y\left(x\right)\right)$

$\left[\left[\left[0\,1\right]\,\left[0\,2y\right]\,\left[0\,y^2\right]\right]\right]$