Table 1 Elements of the Lie algebra \(\mathfrak g\) and the associated Lie group G
Element \(\varvec{{\mathbf {Z}}}_{\varvec{I,r}} \varvec{\in {\mathfrak g}}\) | Element \(\varvec{\exp \left( {{\mathbf {Z}}_{I,r}}\right) \in G}\) | Action of \(\varvec{\exp \left( {{\mathbf {Z}}_{I,r}}\right) }\) |
|---|---|---|
\({\mathbf {Z}}_{1,1} := \begin{pmatrix} 1 &{} - {\mathbf {x}}_0^1 \\ 0 &{} 0\end{pmatrix}\) | \(\exp \left( {{\mathbf {Z}}_{1,1}}\right) = \begin{pmatrix} \mathrm{{e}} &{} ({1-\mathrm{{e}}}) {\mathbf {x}}_0^1 \\ 0 &{} 1\end{pmatrix}\) | Dilatation centered on the position of node 1 |
\({\mathbf {Z}}_{1,2} := \begin{pmatrix} \mathrm{i} &{} -\mathrm{i}{\mathbf {x}}_0^1 \\ 0 &{} 0\end{pmatrix}\) | \(\exp \left( {{\mathbf {Z}}_{1,2}}\right) = \begin{pmatrix} \mathrm{{e}}^{\mathrm{i}} &{} ({1-\mathrm{{e}}^{\mathrm{i}}}) {\mathbf {x}}_0^1 \\ 0 &{} 1\end{pmatrix}\) | Rotation around the position of node 1 |
\({\mathbf {Z}}_{2,1} := \begin{pmatrix} 1 &{} -{\mathbf {x}}_0^2 \\ 0 &{} 0\end{pmatrix}\) | \(\exp \left( {{\mathbf {Z}}_{2,1}}\right) = \begin{pmatrix} \mathrm{{e}} &{} ({1-\mathrm{{e}}}) {\mathbf {x}}_0^2 \\ 0 &{} 1\end{pmatrix}\) | Dilatation centered on the position of node 2 |
\({\mathbf {Z}}_{2,2} := \begin{pmatrix} \mathrm{i} &{} -\mathrm{i}{\mathbf {x}}_0^2 \\ 0 &{} 0\end{pmatrix}\) | \(\exp \left( {{\mathbf {Z}}_{2,2}}\right) = \begin{pmatrix} \mathrm{{e}}^{\mathrm{i}} &{} ({1-\mathrm{{e}}^{\mathrm{i}}}) {\mathbf {x}}_0^2 \\ 0 &{} 1\end{pmatrix}\) | Rotation around the position of node 2 |