solve group - Maple Help

DEtools

 solve_group
 represent a Lie Algebra of symmetry generators in terms of derived algebras

 Calling Sequence solve_group(G, y(x))

Parameters

 G - list of symmetry generators y(x) - dependent and independent variables

Description

 • 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](..).

Examples

 > $\mathrm{with}\left(\mathrm{DEtools}\right):$
 > $G≔\left[\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right]\right]$
 ${G}{≔}\left[\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]\right]$ (1)
 > $\mathrm{solve_group}\left(G,y\left(x\right)\right)$
 $\left[\left[\right]{,}\left[\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]\right]\right]$ (2)
 > $\mathrm{G20}≔\left[\left[0,1\right],\left[1,0\right]\right]$
 ${\mathrm{G20}}{≔}\left[\left[{0}{,}{1}\right]{,}\left[{1}{,}{0}\right]\right]$ (3)
 > $\mathrm{Xcommutator}\left(\mathrm{op}\left(\mathrm{G20}\right),y\left(x\right)\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{0}\right]$ (4)
 > $\mathrm{solve_group}\left(\mathrm{G20},y\left(x\right)\right)$
 $\left[\left[\right]{,}\left[\left[{0}{,}{1}\right]{,}\left[{1}{,}{0}\right]\right]\right]$ (5)
 > $\mathrm{G21}≔\left[\left[0,1\right],\left[0,y\right]\right]$
 ${\mathrm{G21}}{≔}\left[\left[{0}{,}{1}\right]{,}\left[{0}{,}{y}\right]\right]$ (6)
 > $\mathrm{Xcommutator}\left(\mathrm{op}\left(\mathrm{G21}\right),y\left(x\right)\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{1}\right]$ (7)
 > $\mathrm{solve_group}\left(\mathrm{G21},y\left(x\right)\right)$
 $\left[\left[\right]{,}\left[\left[{0}{,}{1}\right]\right]{,}\left[\left[{0}{,}{y}\right]\right]\right]$ (8)
 > $G≔\left[\left[1,0\right],\left[0,1\right],\left[\mathrm{exp}\left(y\right),0\right]\right]$
 ${G}{≔}\left[\left[{1}{,}{0}\right]{,}\left[{0}{,}{1}\right]{,}\left[{{ⅇ}}^{{y}}{,}{0}\right]\right]$ (9)
 > $\mathrm{Xcommutator}\left(G\left[1\right],G\left[2\right],y\left(x\right)\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{0}\right]$ (10)
 > $\mathrm{Xcommutator}\left(G\left[1\right],G\left[3\right],y\left(x\right)\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{0}\right]$ (11)
 > $\mathrm{Xcommutator}\left(G\left[2\right],G\left[3\right],y\left(x\right)\right)$
 $\left[{\mathrm{_ξ}}{=}{{ⅇ}}^{{y}}{,}{\mathrm{_η}}{=}{0}\right]$ (12)
 > $\mathrm{solve_group}\left(G,y\left(x\right)\right)$
 $\left[\left[\right]{,}\left[\left[{{ⅇ}}^{{y}}{,}{0}\right]\right]{,}\left[\left[{1}{,}{0}\right]{,}\left[{0}{,}{1}\right]\right]\right]$ (13)
 > $\mathrm{SL2}≔\left[\left[0,1\right],\left[0,y\right],\left[0,{y}^{2}\right]\right]$
 ${\mathrm{SL2}}{≔}\left[\left[{0}{,}{1}\right]{,}\left[{0}{,}{y}\right]{,}\left[{0}{,}{{y}}^{{2}}\right]\right]$ (14)
 > $\mathrm{Xcommutator}\left(\mathrm{SL2}\left[1\right],\mathrm{SL2}\left[2\right],y\left(x\right)\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{1}\right]$ (15)
 > $\mathrm{Xcommutator}\left(\mathrm{SL2}\left[1\right],\mathrm{SL2}\left[3\right],y\left(x\right)\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{2}{}{y}\right]$ (16)
 > $\mathrm{Xcommutator}\left(\mathrm{SL2}\left[2\right],\mathrm{SL2}\left[3\right],y\left(x\right)\right)$
 $\left[{\mathrm{_ξ}}{=}{0}{,}{\mathrm{_η}}{=}{{y}}^{{2}}\right]$ (17)
 > $\mathrm{solve_group}\left(\mathrm{SL2},y\left(x\right)\right)$
 $\left[\left[\left[{0}{,}{1}\right]{,}\left[{0}{,}{2}{}{y}\right]{,}\left[{0}{,}{{y}}^{{2}}\right]\right]\right]$ (18)