Let $A \= \left\{A_{1 }\, A_2\, ..\. A_p\right\}$be a list of $p$square matrices. The set of matrices group $A$defines a matrix Lie algebra if it closed with respect to the matrix commutator, that is, for all $i\,j \= 1..\. p$ there are constants $C_{\mathrm{ij} }^k$such that: $\left[A_{i\,} A_j\right] \= A_i A_j-A_j{A}_{i }\= C_{\mathrm{ij} }^k A_k \.$

A matrix $B\left(x^1\, x^2 \, ..\. \, x^p\right)$, which is a function of $p$ variables $\left\{ x^1\, x^2 \, ..\. \, {x^p}_{ }\right\}$, defines a matrix group if :

If $B\left(x^1\, x^2 \, ..\. \, x^p\right)$ is a matrix group, then the partial derivatives  $A_{i }\= \left(\frac{\partial B}{\partial x^i}\right)$$\left(x_0^1\, x_0^2 \, ..\. \, x_0^p\right)$ define a matrix algebra.

 The first calling sequence MatrixGroup(A, vars) calculates the matrix group defined by the matrix algebra $A \= \left\{A_{1 }\, A_2\, ..\. \,A_p\right\}$from the products of  matrix exponentials $\exp \left(t_{i }{A}_i\right)\.$Simple heuristics are used to relabel the group parameters$t_i$in terms of the variables $\mathrm{vars} \=$$\left\{ x^1\, x^2 \, ..\. \, {x^p}_{ }\right\}$in order to simplify the final expressions for the components of the matrix $B\left(x^1\, x^2 \, ..\. \, x^p\right)$. These heuristics work best if the basis for matrix algebra $A$ is given, to the maximal extent possible, in terms of strictly upper triangular matrices, strictly lower triangular matrices, and diagonal matrices. The matrix exponentials are computed using MatrixExponential.

 For the given matrix group $B\left(x^1\, x^2 \, ..\. \, x^p\right)$, the left invariant and right invariant Maurer-Cartan forms are the matrices of 1-forms defined by

The second calling sequence MatrixGroup(Omega, LR) returns the matrix group $B\left(x^1\, x^2 \, ..\. \, x^p\right)$which satisfies the equations (*). The argument Omega is interpreted to be the left or right invariant Maurer-Cartan forms according to the value of the string LR. The equations (*) are integrated using pdsolve. If the integrability conditions on Omega are not satisfied, the pdsolve command will issues a warning and MatrixGroup will return NULL.

The command GroupAction is part of the DifferentialGeometry:-GroupActions package.  It can be used in the form MatrixGroup(...) only after executing the commands with(DifferentialGeometry) and with(GroupActions), but can always be used by executing DifferentialGeometry:-GroupActions:-MatrixGroup(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{GroupActions}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$$\mathrm{with}\left(\mathrm{Library}\right)\:$$\mathrm{with}\left(\mathrm{Tools}\right)\:$

$A≔\left[\mathrm{Matrix}\left(\left[\left[1\,0\right]\,\left[0\,1\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,1\right]\,\left[0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,0\right]\,\left[0\,1\right]\right]\right)\right]$

$\mathrm{LieAlgebraData}\left(A\right)$

$\left[\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e2}\right]$

$B,\mathrm{id}≔\mathrm{MatrixGroup}\left(A\,\left[x\,y\,z\right]\right)$

$C≔\left[\mathrm{seq}\left(\mathrm{map}\left(\mathrm{diff}\,B\,v\right)\,v=\left[x\,y\,z\right]\right)\right]$

$\mathrm{GetComponents}\left(C\,A\,\mathrm{trueorfalse}=''on''\right)$

$\mathrm{true}$

$A≔\left[\mathrm{Matrix}\left(\left[\left[1\,0\right]\,\left[0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,1\right]\,\left[0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,0\right]\,\left[1\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,0\right]\,\left[0\,1\right]\right]\right)\right]$

$B,\mathrm{id}≔\mathrm{MatrixGroup}\left(A\,\left[a\,b\,c\,d\right]\right)$

$X≔\mathrm{seq}\left(\mathrm{eval}\left(\mathrm{LinearAlgebra}:-\mathrm{MatrixExponential}\left(t\Vert iA\left[i\right]\right)\right)\,i=1..4\right)$

$X\left[1\right]·X\left[2\right]·X\left[3\right]·X\left[4\right]$

$A≔\left[\mathrm{Matrix}\left(\left[\left[1\,0\right]\,\left[0\,-1\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,1\right]\,\left[0\,0\right]\right]\right)\,\mathrm{Matrix}\left(\left[\left[0\,0\right]\,\left[1\,0\right]\right]\right)\right]$

$B,\mathrm{id}≔\mathrm{MatrixGroup}\left(A\,\left[x\,y\,z\right]\right)$

$\mathrm{LinearAlgebra}:-\mathrm{Determinant}\left(B\right)$

$1$

$\mathrm{LD}≔\mathrm{Retrieve}\left(''Winternitz''\,1\,\left[5\,37\right]\,\mathrm{alg}\right)$

$\mathrm{LD}:=\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\=2\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=-\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; alg}$

$A≔\mathrm{Adjoint}\left(\right)$

$\mathrm{MatrixGroup}\left(A\,\left[u\,v\,x\,y\,z\right]\right)$

$\mathrm{LD}≔\mathrm{Retrieve}\left(''Winternitz''\,1\,\left[5\,5\right]\,\mathrm{alg}\right)$

$\mathrm{LD}:=\left[\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e1}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; alg}$

$A≔\mathrm{Derivations}\left(''Full''\right)$

$\mathrm{MatrixGroup}\left(A\,\left[\mathrm{seq}\left(x\Vert i\,i=1..\mathrm{nops}\left(A\right)\right)\right]\right)$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$\mathrm{\Omega }≔\mathrm{Matrix}\left(2\,2\,\left[\mathrm{evalDG}\left(\frac{1}{x}\mathrm{dx}\right)\,\mathrm{evalDG}\left(-\frac{z\mathrm{dy}}{xy}+\frac{\mathrm{dz}}{x}\right)\,0\&mult\mathrm{dx}\,\mathrm{evalDG}\left(\frac{\mathrm{dy}}{y}\right)\right]\right)$

$\mathrm{ExteriorDerivative}\left(\mathrm{\Omega }\right)\&MatrixPlus\left(\mathrm{\Omega }\&MatrixWedge\mathrm{\Omega }\right)$

$\mathrm{MatrixGroup}\left(\mathrm{\Omega }\,''Left''\,\left[x=1\,y=1\,z=0\right]\right)$