ApplyHomomorphism($\mathbf{\φ}$, T) will apply the transformation Phi to the vector, form or tensor T and return an object of the same type. The precise evaluation rules for ApplyHomomorphism depend upon the specific properties of T and whether or not Phi is invertible. The details are as follows.

Applied to tensors, the command ApplyHomomorphism acts as a ring homomorphism, that is, ApplyHomomorphism($\mathbf{\φ}$, T$\mathbf{\otimes }$S) = ApplyHomomorphism($\mathbf{\φ}$, T)$\mathbf{\otimes }$ApplyHomomorphism($\mathbf{\φ}$, S).

CASE 1. T is a vector in the domain algebra $\mathrm{\𝔤}\mathit{ }$of $\mathbf{\φ}$. In this case ApplyHomomorphism(Phi, T) simply applies the linear transformation $\mathbf{\φ}$ to the vector T and the result is a vector in the range algebra $\mathrm{\𝔥}\mathit{ }$of the transformation $\mathbf{\φ}$.

CASE 2. T is a $p$-form on the range algebra $\mathrm{\𝔥}$ of transformation $\mathbf{\φ}$.In this case ApplyHomomorphism($\mathbf{\φ}$, T) simply applies the pullback of the linear transformation $\mathbf{\φ}$ to the $p$-form T and the result is a $p$-form in the domain $\mathrm{\𝔤}$ of $\mathbf{\φ}$.

CASE 3. T is a tensor on $\mathrm{\𝔤}$ and $\mathbf{\φ}$ is an invertible linear transformation. Then ApplyHomomorphism($\mathbf{\φ}$, T) is the tensor on the range algebra $\mathrm{\𝔥}$ obtained by the pushforward by $\mathrm{\φ}$of the contravariant components of T and the pullback of the covariant components of T by the inverse of $\mathbf{\φ}$.

CASE 4. T is a tensor on $\mathrm{\𝔥}\mathit{ }$ and $\mathbf{\φ}$ is an invertible linear transformation. Then ApplyHomomorphism($\mathbf{\φ}$, T) is the tensor on the domain algebra $\mathrm{\𝔤}\mathit{ }$obtained by the pushforward of the contravariant components of T by the inverse of $\mathbf{\φ}$ and the pullback of the covariant components of T by $\mathbf{\φ}$.

CASE 5. T is a tensor on $\mathrm{\𝔤}\mathit{ }$and $\mathbf{\φ}$ is not invertible. Then T must be a contravariant tensor (that is, a tensor products of vectors) in which case ApplyHomomorphism($\mathbf{\φ}$, T) is the contravariant tensor defined on the range algebra $\mathrm{\𝔥}$ and obtained by the pushforward of Phi acting on vectors in $\mathrm{\𝔤}$.

When $\mathrm{\𝔥} \=\mathrm{\𝔤}$, Case 4 takes precedence over Case 5. Alternatively ApplyHomomorphism can be forced to use Case 4 or Case 5 with the third optional argument "domain" or "range".

CASE 6. T is a tensor on $\mathrm{\𝔥}$ and $\mathbf{\φ}$ is not invertible.  Then T must be a covariant tensor (that is, a tensor product of 1-forms) in which case ApplyHomomorphism($\mathbf{\φ}$, T) is the covariant tensor defined on the domain algebra g and obtained by the pullback of $\mathbf{\φ}$ acting on 1-forms in $h$.

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

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

$\mathrm{LC1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[4\right]\right]\,\left[\left[\left[1\,4\,1\right]\,1\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,4\,2\right]\,1\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{LC1}\,\left[x\right]\,\left[\mathrm{\alpha }\right]\right)\:$

$\mathrm{LC2}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg2}\,\left[4\right]\right]\,\left[\left[\left[1\,4\,1\right]\,1\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,4\,2\right]\,1\right]\right]\right]\right)\:$

$\mathrm{DGsetup}\left(\mathrm{LC2}\,\left[y\right]\,\left[\mathrm{\beta }\right]\right)\:$

$\mathrm{print}\left(\mathrm{MultiplicationTable}\left(\mathrm{Alg1}\,''LieBracket''\right)\,\mathrm{MultiplicationTable}\left(\mathrm{Alg2}\,''LieBracket''\right)\right)$

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

$A≔\mathrm{AdjointExp}\left(r\mathrm{x1}+s\mathrm{x2}+t\mathrm{x3}\right)\:$

$\mathrm{\Phi }≔\mathrm{Transformation}\left(\mathrm{Alg1}\,\mathrm{Alg2}\,A\right)$

$\mathrm{\Φ}:=\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{Alg1}\,0\right]\,\left[\mathrm{Alg2}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{cccc}1& -t& s& -\frac{ts}{2}+r\\ 0& 1& 0& s\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\right]\right]\,\left[\left[\mathrm{\_z1}-t\mathrm{\_z2}+s\mathrm{\_z3}+\left(-\frac{ts}{2}+r\right)\mathrm{\_z4}\,\mathrm{\_z5}\right]\,\left[s\mathrm{\_z4}+\mathrm{\_z2}\,\mathrm{\_z6}\right]\,\left[\mathrm{\_z3}\,\mathrm{\_z7}\right]\,\left[\mathrm{\_z4}\,\mathrm{\_z8}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{Alg1}\,0\right]\,\left[\mathrm{Alg2}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{cccc}1& -t& s& -\frac{ts}{2}+r\\ 0& 1& 0& s\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\right]\right]\,\left[\left[\mathrm{\_z1}-t\mathrm{\_z2}+s\mathrm{\_z3}+\left(-\frac{ts}{2}+r\right)\mathrm{\_z4}\,\mathrm{\_z5}\right]\,\left[s\mathrm{\_z4}+\mathrm{\_z2}\,\mathrm{\_z6}\right]\,\left[\mathrm{\_z3}\,\mathrm{\_z7}\right]\,\left[\mathrm{\_z4}\,\mathrm{\_z8}\right]\right]\right]\right)$

$\mathrm{Vectors}≔\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\:$

$A,\mathrm{map2}\left(\mathrm{ApplyHomomorphism}\,\mathrm{\Phi }\,\mathrm{Vectors}\right)$

$\left[\begin{array}{cccc}1& -t& s& -\frac{1}{2}ts\+r\\ 0& 1& 0& s\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\,\left[\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Alg2}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Alg2}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Alg2}\,\left[\right]\right]\,\left[\left[\left[1\right]\,-t\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Alg2}\,\left[\right]\right]\,\left[\left[\left[1\right]\,-t\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Alg2}\,\left[\right]\right]\,\left[\left[\left[1\right]\,s\right]\,\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Alg2}\,\left[\right]\right]\,\left[\left[\left[1\right]\,s\right]\,\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Alg2}\,\left[\right]\right]\,\left[\left[\left[1\right]\,-\frac{ts}{2}+r\right]\,\left[\left[2\right]\,s\right]\,\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{Alg2}\,\left[\right]\right]\,\left[\left[\left[1\right]\,-\frac{ts}{2}+r\right]\,\left[\left[2\right]\,s\right]\,\left[\left[4\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{Forms}≔\left[\mathrm{\β1}\,\mathrm{\β2}\,\mathrm{\β3}\,\mathrm{\β4}\right]\:$

$\mathrm{Atr}≔\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(A\right)\:$

$\mathrm{Atr},\mathrm{map2}\left(\mathrm{ApplyHomomorphism}\,\mathrm{\Phi }\,\mathrm{Forms}\right)$

$\left[\begin{array}{cccc}1& 0& 0& 0\\ -t& 1& 0& 0\\ s& 0& 1& 0\\ -\frac{1}{2}ts\+r& s& 0& 1\end{array}\right]\,\left[\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Alg1}\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,-t\right]\,\left[\left[3\right]\,s\right]\,\left[\left[4\right]\,-\frac{ts}{2}+r\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Alg1}\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,-t\right]\,\left[\left[3\right]\,s\right]\,\left[\left[4\right]\,-\frac{ts}{2}+r\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Alg1}\,1\right]\,\left[\left[\left[2\right]\,1\right]\,\left[\left[4\right]\,s\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Alg1}\,1\right]\,\left[\left[\left[2\right]\,1\right]\,\left[\left[4\right]\,s\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Alg1}\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Alg1}\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Alg1}\,1\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{Alg1}\,1\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{CovariantTensors}≔\mathrm{map}\left(\mathrm{convert}\,\left[\mathrm{\α1}\,\mathrm{\α2}\,\mathrm{\α3}\,\mathrm{\α4}\right]\,\mathrm{DGtensor}\right)$

$\mathrm{CovariantTensors}:=\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{Aintr}≔\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(A\right)\right)\:$

$\mathrm{Aintr},\mathrm{map2}\left(\mathrm{ApplyHomomorphism}\,\mathrm{\Phi }\,\mathrm{CovariantTensors}\right)$

$\left[\begin{array}{cccc}1& 0& 0& 0\\ t& 1& 0& 0\\ -s& 0& 1& 0\\ -\frac{1}{2}ts-r& -s& 0& 1\end{array}\right]\,\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,t\right]\,\left[\left[3\right]\,-s\right]\,\left[\left[4\right]\,-\frac{ts}{2}-r\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,t\right]\,\left[\left[3\right]\,-s\right]\,\left[\left[4\right]\,-\frac{ts}{2}-r\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\right]\,1\right]\,\left[\left[4\right]\,-s\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\right]\,1\right]\,\left[\left[4\right]\,-s\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[4\right]\,1\right]\right]\right]\right)\right]$

$\mathrm{ContravariantTensors}≔\mathrm{map}\left(\mathrm{convert}\,\left[\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right]\,\mathrm{DGtensor}\right)\:$

$\mathrm{Ain}≔\mathrm{LinearAlgebra}:-\mathrm{MatrixInverse}\left(A\right)\:$

$\mathrm{Ain},\mathrm{map2}\left(\mathrm{ApplyHomomorphism}\,\mathrm{\Phi }\,\mathrm{ContravariantTensors}\right)$

$\left[\begin{array}{cccc}1& t& -s& -\frac{1}{2}ts-r\\ 0& 1& 0& -s\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\,\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,t\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,t\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,-s\right]\,\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,-s\right]\,\left[\left[3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,-\frac{ts}{2}-r\right]\,\left[\left[2\right]\,-s\right]\,\left[\left[4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,-\frac{ts}{2}-r\right]\,\left[\left[2\right]\,-s\right]\,\left[\left[4\right]\,1\right]\right]\right]\right)\right]$

$T≔\mathrm{\α2}\&tensor\mathrm{x3}$

$T:=\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,3\right]\,1\right]\right]\right]\right)$

$\mathrm{ApplyHomomorphism}\left(\mathrm{\Phi }\,T\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\right]\,s\right]\,\left[\left[2\,3\right]\,1\right]\,\left[\left[4\,1\right]\,-s^2\right]\,\left[\left[4\,3\right]\,-s\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\right]\,s\right]\,\left[\left[2\,3\right]\,1\right]\,\left[\left[4\,1\right]\,-s^2\right]\,\left[\left[4\,3\right]\,-s\right]\right]\right]\right)$

$T≔\mathrm{\β2}\&tensor\mathrm{y3}$

$T:=\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,3\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg2}\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,3\right]\,1\right]\right]\right]\right)$

$\mathrm{ApplyHomomorphism}\left(\mathrm{\Phi }\,T\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\right]\,-s\right]\,\left[\left[2\,3\right]\,1\right]\,\left[\left[4\,1\right]\,-s^2\right]\,\left[\left[4\,3\right]\,s\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{Alg1}\,\left[\left[''cov\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,1\right]\,-s\right]\,\left[\left[2\,3\right]\,1\right]\,\left[\left[4\,1\right]\,-s^2\right]\,\left[\left[4\,3\right]\,s\right]\right]\right]\right)$