If $\mathrm{\ρ}\:\mathrm{\𝔤}\mathit{ }\to \mathrm{gl}\left(V\right)$is a representation and $\mathrm{\𝔥}$ is a subalgebra of $\mathrm{\𝔤}$ , then the restriction of $\mathrm{\ρ}$ to $\mathrm{\𝔥}$ is the representation $\mathrm{\φ}\:\mathrm{\𝔥}\mathit{ }\mathit{\to }\mathit{ }\mathrm{gl}\left(V\right)$defined by $\mathrm{\φ}\left(x\right)\left(Y\right)$ =$\mathrm{\rho }\left(x\right)\left(Y\right)$, where $x\mathit{ }\in \mathrm{\𝔥}$ and $Y \in V$. 

The command RestrictedRepresentation(rho, alg, H) returns the restriction of the representation $\mathrm{\ρ}$to the subalgebra defined by the vectors in the list $H\.$$$The subalgebra defined by the vectors $H$must be initialized as a Lie algebra in its own right with the name alg.

If the basis $\left\{e_1\, e_2 \,e_3\, ..\. \, e_{n }\right\}$for $\mathrm{\𝔤}$ is adapted to the subalgebra defined by $H$in the sense that $H \= \[$$e_1\, e_2 \,..\. \,e_p \]\,$then the list $H$need not be specified in the calling sequence for RestrictedRepresentation.

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

$L≔\mathrm{Retrieve}\left(''Winternitz''\,1\,\left[4\,7\right]\,\mathrm{Alg}\right)$

$L:=\left[\left[\mathrm{e1}\,\mathrm{e4}\right]\=2\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e4}\right]\=\mathrm{e2}\,\left[\mathrm{e3}\,\mathrm{e4}\right]\=\mathrm{e2}\+\mathrm{e3}\right]$

$\mathrm{DGsetup}\left(L\right)\:$

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

$\mathrm{\rho }≔\mathrm{Adjoint}\left(\mathrm{Alg}\,\mathrm{representationspace}=V\right)$

$\mathrm{H1}≔\left[\mathrm{e1}\,\mathrm{e2}\right]$

$\mathrm{H1}:=\left[\mathrm{e1}\,\mathrm{e2}\right]$

$\mathrm{L1}≔\mathrm{LieAlgebraData}\left(\mathrm{H1}\,\mathrm{Alg1}\right)$

$\mathrm{L1}:=\left[\right]$

$\mathrm{DGsetup}\left(\mathrm{L1}\,\left['P'\right]\,\left['p'\right]\right)$

$\mathrm{Lie\; algebra:\; Alg1}$

$\mathrm{\ρ1}≔\mathrm{RestrictedRepresentation}\left(\mathrm{\rho }\,\mathrm{Alg1}\right)$

$\mathrm{Query}\left(\mathrm{\ρ1}\,''Representation''\right)$

$\mathrm{true}$

$\mathrm{H2}≔\left[\mathrm{e4}+\mathrm{e2}\,\mathrm{e2}\right]$

$\mathrm{H2}:=\left[\mathrm{e4}\+\mathrm{e2}\,\mathrm{e2}\right]$

$\mathrm{L2}≔\mathrm{LieAlgebraData}\left(\mathrm{H2}\,\mathrm{Alg2}\right)$

$\mathrm{L2}:=\left[\left[\mathrm{e1}\,\mathrm{e2}\right]\=-\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{L2}\,\left['Q'\right]\,\left['q'\right]\right)$

$\mathrm{Lie\; algebra:\; Alg2}$

$\mathrm{\ρ2}≔\mathrm{RestrictedRepresentation}\left(\mathrm{\rho }\,\mathrm{Alg2}\,\mathrm{H2}\right)$

$\mathrm{Query}\left(\mathrm{\ρ2}\,''Representation''\right)$

$\mathrm{true}$