Let $G$ be a Lie group with multiplication * and identity $e$. In the DifferentialGeometry package a Lie group is represented by a module with exports Frame, Identity, LeftMultiplication, RightMultiplication, and Inverse. The Frame export returns the name given to the underlying manifold used to define coordinates on the Lie group. The export Identity gives the coordinate list for the identity point $e$. LeftMultiplication and RightMultiplication are procedures which accept the coordinates of a point $a$ as arguments and return the transformations from $G$ to $G$ defining left multiplication $x\to a\*x$ and right multiplication $x\to x\*a$. Inverse returns the transformation from $G$ to $G$ defined by $x\to x^{-1}$.

The first calling sequence computes the above Lie group module for a matrix group, that is, an $r$-parameter set of matrices containing the identity and closed under matrix multiplication and matrix inversion. The parameters appearing in the matrix $M$ must match the coordinates used to define the Lie group $G\.$

The second calling sequence computes the Lie group module for a given $r$-parameter transformation group, defined by the transformation T. The parameters appearing in the transformation T must match the coordinates used to define the Lie group $G$.

In the third calling sequence the transformation in taken to be the transformation defining either the left  ($x\to a\*x$)  or right ( $x\to x\*a$) multiplication for the Lie group $G$. The coordinates of the generic point $a\in G$are specified by the list vars.

The fourth calling sequence implements Lie's second and third theorems and constructs a global Lie group $G$ for any solvable Lie algebra Alg.

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

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

$X≔\mathrm{Matrix}\left(\left[\left[x\,y\right]\,\left[0\,z\right]\right]\right)$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\mathrm{G1}\right)$

$\mathrm{frame\; name:\; G1}$

$\mathrm{LG1}≔\mathrm{LieGroup}\left(X\,\mathrm{G1}\right)$

$\mathrm{LG1}:=\mathbf{module}\left(\right)\mathbf{export}\mathrm{Frame}\,\mathrm{Identity}\,\mathrm{LeftMultiplication}\,\mathrm{RightMultiplication}\,\mathrm{Inverse}\;\mathbf{end\; module}$

$\mathrm{LG1}:-\mathrm{Frame}$

$\mathrm{G1}$

$\mathrm{LG1}\left[\mathrm{Identity}\right]$

$\left[x\=1\,y\=0\,z\=1\right]$

$\mathrm{muL}≔\mathrm{LG1}:-\mathrm{LeftMultiplication}\left(\left[a\,b\,c\right]\right)$

$\mathrm{muL}:=\left[x\=ax\,y\=ay\+bz\,z\=cz\right]$

$\mathrm{muR}≔\mathrm{LG1}:-\mathrm{RightMultiplication}\left(\left[a\,b\,c\right]\right)$

$\mathrm{muR}:=\left[x\=ax\,y\=bx\+cy\,z\=cz\right]$

$\mathrm{muInv}≔\mathrm{LG1}:-\mathrm{Inverse}$

$\mathrm{muInv}:=\left[x\=\frac{1}{x}\,y\=-\frac{y}{xz}\,z\=\frac{1}{z}\right]$

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

$L≔A·X$

$R≔X·A$

$Y≔\frac{1}{X}$

$\mathrm{ApplyTransformation}\left(\mathrm{muL}\,\mathrm{ApplyTransformation}\left(\mathrm{muInv}\,\left[a\,b\,c\right]\right)\right)$

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

$\mathrm{InvariantVectorsAndForms}\left(\mathrm{LG1}\right)$

$\left[x\mathrm{D\_x}\,x\mathrm{D\_y}\,\mathrm{D\_y}y\+\mathrm{D\_z}z\right]\,\left[\frac{\mathrm{dx}}{x}\,\frac{\mathrm{dy}}{x}-\frac{y\mathrm{dz}}{xz}\,\frac{\mathrm{dz}}{z}\right]\,\left[\mathrm{D\_x}x\+\mathrm{D\_y}y\,z\mathrm{D\_y}\,z\mathrm{D\_z}\right]\,\left[\frac{\mathrm{dx}}{x}\,-\frac{y\mathrm{dx}}{xz}\+\frac{\mathrm{dy}}{z}\,\frac{\mathrm{dz}}{z}\right]$

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

$\mathrm{frame\; name:\; M}$

$\mathrm{T2}≔\mathrm{Transformation}\left(M\,M\,\left[x=ax+by+c\,y=dy+e\right]\right)$

$\mathrm{T2}:=\left[x\=ax\+by\+c\,y\=dy\+e\right]$

$\mathrm{DGsetup}\left(\left[a\,b\,c\,d\,e\right]\,\mathrm{G2}\right)$

$\mathrm{frame\; name:\; G2}$

$\mathrm{LG2}≔\mathrm{LieGroup}\left(\mathrm{T2}\,\mathrm{G2}\right)$

$\mathrm{LG2}:=\mathbf{module}\left(\right)\mathbf{export}\mathrm{Frame}\,\mathrm{Identity}\,\mathrm{LeftMultiplication}\,\mathrm{RightMultiplication}\,\mathrm{Inverse}\;\mathbf{end\; module}$

$\mathrm{LG2}:-\mathrm{LeftMultiplication}\left(\left[\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,\mathrm{\epsilon }\right]\right)$

$\left[a\=a\mathrm{\α}\,b\=b\mathrm{\α}\+d\mathrm{\β}\,c\=c\mathrm{\α}\+e\mathrm{\β}\+\mathrm{\γ}\,d\=\mathrm{\δ}d\,e\=e\mathrm{\δ}\+\mathrm{\ϵ}\right]$

$\mathrm{DGsetup}\left(\left[a\,b\,c\,d\,e\right]\,\mathrm{G2}\right)$

$\mathrm{frame\; name:\; G2}$

$\mathrm{LG2}≔\mathrm{LieGroup}\left(\mathrm{T2}\,\mathrm{G2}\right)$

$\mathrm{LG2}:=\mathbf{module}\left(\right)\mathbf{export}\mathrm{Frame}\,\mathrm{Identity}\,\mathrm{LeftMultiplication}\,\mathrm{RightMultiplication}\,\mathrm{Inverse}\;\mathbf{end\; module}$

$\mathrm{LG2}:-\mathrm{LeftMultiplication}\left(\left[\mathrm{\alpha }\,\mathrm{\beta }\,\mathrm{\gamma }\,\mathrm{\delta }\,\mathrm{\epsilon }\right]\right)$

$\left[a\=a\mathrm{\α}\,b\=b\mathrm{\α}\+d\mathrm{\β}\,c\=c\mathrm{\α}\+e\mathrm{\β}\+\mathrm{\γ}\,d\=\mathrm{\δ}d\,e\=e\mathrm{\δ}\+\mathrm{\ϵ}\right]$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\mathrm{G3}\right)$

$\mathrm{frame\; name:\; G3}$

$\mathrm{T3}≔\mathrm{Transformation}\left(\mathrm{G3}\,\mathrm{G3}\,\left[x=ax\,y=ay+bz\,z=cz\right]\right)$

$\mathrm{T3}:=\left[x\=ax\,y\=ay\+bz\,z\=cz\right]$

$\mathrm{LG3L}≔\mathrm{LieGroup}\left(\mathrm{T3}\,\left[a\,b\,c\right]\right)$

$\mathrm{LG3L}:=\mathbf{module}\left(\right)\mathbf{export}\mathrm{Frame}\,\mathrm{Identity}\,\mathrm{LeftMultiplication}\,\mathrm{RightMultiplication}\,\mathrm{Inverse}\;\mathbf{end\; module}$

$\mathrm{S3}≔\mathrm{LG3L}:-\mathrm{RightMultiplication}\left(\left[a\,b\,c\right]\right)$

$\mathrm{S3}:=\left[x\=ax\,y\=bx\+cy\,z\=cz\right]$

$\mathrm{LG3R}≔\mathrm{LieGroup}\left(\mathrm{S3}\,\left[a\,b\,c\right]\,\mathrm{multiplication}=''right''\right)$

$\mathrm{LG3R}:=\mathbf{module}\left(\right)\mathbf{export}\mathrm{Frame}\,\mathrm{Identity}\,\mathrm{LeftMultiplication}\,\mathrm{RightMultiplication}\,\mathrm{Inverse}\;\mathbf{end\; module}$

$\mathrm{LG3R}:-\mathrm{RightMultiplication}\left(\left[a\,b\,c\right]\right)$

$\left[x\=ax\,y\=bx\+cy\,z\=cz\right]$

$\mathrm{LG3R}:-\mathrm{LeftMultiplication}\left(\left[a\,b\,c\right]\right)$

$\left[x\=ax\,y\=ay\+bz\,z\=cz\right]$

$\mathrm{LA}≔\mathrm{Library}:-\mathrm{Retrieve}\left(''Winternitz''\,1\,\left[4\,8\right]\,\mathrm{alg}\right)$

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

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

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

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\,\mathrm{G4}\right)$

$\mathrm{frame\; name:\; G4}$

$\mathrm{LG4}≔\mathrm{LieGroup}\left(\mathrm{alg}\,\mathrm{G4}\right)$

$\mathrm{LG4}:=\mathbf{module}\left(\right)\mathbf{export}\mathrm{Frame}\,\mathrm{Identity}\,\mathrm{LeftMultiplication}\,\mathrm{RightMultiplication}\,\mathrm{Inverse}\;\mathbf{end\; module}$

$\mathrm{LG4}:-\mathrm{LeftMultiplication}\left(\left[\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right]\right)$

$\left[\mathrm{x1}\=\mathrm{y1}-{\ⅇ}^{-\mathrm{y4}}\mathrm{x3}\mathrm{x4}\mathrm{y2}-{\ⅇ}^{-\mathrm{y4}}\mathrm{x3}\mathrm{y2}\mathrm{y4}-{\ⅇ}^{-\mathrm{y4}}\mathrm{x3}\mathrm{y2}-\mathrm{y2}\mathrm{y3}\mathrm{x4}-\mathrm{x2}{\ⅇ}^{\mathrm{y4}}\mathrm{y3}\mathrm{x4}-\mathrm{x2}{\ⅇ}^{\mathrm{y4}}\mathrm{y3}\mathrm{y4}-\mathrm{y4}\mathrm{x2}\mathrm{x3}\+\mathrm{x1}\,\mathrm{x2}\=\mathrm{y2}\+\mathrm{x2}{\ⅇ}^{\mathrm{y4}}\,\mathrm{x3}\=\mathrm{y3}\+\mathrm{x3}{\ⅇ}^{-\mathrm{y4}}\,\mathrm{x4}\=\mathrm{x4}\+\mathrm{y4}\right]$

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\,\mathrm{G4}\right)$

$\mathrm{frame\; name:\; G4}$

$\mathrm{LG4}≔\mathrm{LieGroup}\left(\mathrm{alg}\,\mathrm{G4}\right)$

$\mathrm{LG4}:=\mathbf{module}\left(\right)\mathbf{export}\mathrm{Frame}\,\mathrm{Identity}\,\mathrm{LeftMultiplication}\,\mathrm{RightMultiplication}\,\mathrm{Inverse}\;\mathbf{end\; module}$

$\mathrm{\Γ4}≔\mathrm{InvariantVectorsAndForms}\left(\mathrm{LG4}\right)\left[3\right]$

$\mathrm{\Γ4}:=\left[\mathrm{D\_x1}\,-\left(\mathrm{x3}\mathrm{x4}\+\mathrm{x3}\right)\mathrm{D\_x1}\+\mathrm{D\_x2}\,-\mathrm{x2}\mathrm{x4}\mathrm{D\_x1}\+\mathrm{D\_x3}\,-\mathrm{x2}\mathrm{x3}\mathrm{D\_x1}\+\mathrm{x2}\mathrm{D\_x2}-\mathrm{x3}\mathrm{D\_x3}\+\mathrm{D\_x4}\right]$

$\mathrm{LieAlgebraData}\left(\mathrm{\Γ4}\right)$

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

$\mathrm{DGsetup}\left(\left[x\right]\,\mathrm{M5}\right)$

$\mathrm{frame\; name:\; M5}$

$\mathrm{T5}≔\mathrm{Transformation}\left(\mathrm{M5}\,\mathrm{M5}\,\left[x=\frac{ax+b}{cx+d}\right]\right)$

$\mathrm{T5}:=\left[x\=\frac{ax\+b}{cx\+d}\right]$

$\mathrm{T5}≔\mathrm{eval}\left(\mathrm{T5}\,\left[d=\frac{1+bc}{a}\right]\right)$

$\mathrm{T5}:=\left[x\=\frac{ax\+b}{cx\+\frac{bc\+1}{a}}\right]$

$\mathrm{DGsetup}\left(\left[a\,b\,c\right]\,\mathrm{G5}\right)$

$\mathrm{frame\; name:\; G5}$

$\mathrm{LG5}≔\mathrm{LieGroup}\left(\mathrm{T5}\,\mathrm{G5}\,\mathrm{identity}=\left[a=1\,b=0\,c=0\right]\right)$

$\mathrm{LG5}:=\mathbf{module}\left(\right)\mathbf{export}\mathrm{Frame}\,\mathrm{Identity}\,\mathrm{LeftMultiplication}\,\mathrm{RightMultiplication}\,\mathrm{Inverse}\;\mathbf{end\; module}$

$\mathrm{LG5}\left[\mathrm{RightMultiplication}\right]\left(\left[r\,s\,t\right]\right)$

$\left[a\=ar\+bt\,b\=\frac{ars\+bst\+b}{r}\,c\=\frac{acr\+bct\+t}{a}\right]$

$X≔\mathrm{Matrix}\left(\left[\left[\cos \left(\mathrm{\theta }\right)\,\sin \left(\mathrm{\theta }\right)\,a\right]\,\left[-\sin \left(\mathrm{\theta }\right)\,\cos \left(\mathrm{\theta }\right)\,b\right]\,\left[0\,0\,1\right]\right]\right)$

$\mathrm{DGsetup}\left(\left[\mathrm{\theta }\,a\,b\right]\,\mathrm{G6}\right)$

$\mathrm{frame\; name:\; G6}$

$\mathrm{Preferences}\left(''simplification''\,x\mapsto \mathrm{simplify}\left(\mathrm{combine}\left(x\right)\right)\right)$

$\mathbf{proc}\left(s\right)...\mathbf{end\; proc}$

$\mathrm{G6}≔\mathrm{LieGroup}\left(X\,\mathrm{G6}\right)$

$\mathrm{G6}:=\mathbf{module}\left(\right)\mathbf{export}\mathrm{Frame}\,\mathrm{Identity}\,\mathrm{LeftMultiplication}\,\mathrm{RightMultiplication}\,\mathrm{Inverse}\;\mathbf{end\; module}$

$\mathrm{G6}\left[\mathrm{LeftMultiplication}\right]\left(\left[\mathrm{\phi }\,c\,d\right]\right)$

$\left[\mathrm{\θ}\=\mathrm{\φ}\+\mathrm{\θ}\,a\=\cos \left(\mathrm{\φ}\right)a\+\sin \left(\mathrm{\φ}\right)b\+c\,b\=-\sin \left(\mathrm{\φ}\right)a\+\cos \left(\mathrm{\φ}\right)b\+d\right]$