Table 2 Elements of the Lie algebra \(\tilde{\mathfrak g}\) and the associated Lie group \(\tilde{G}\)
Element \(\varvec{{\tilde{\mathbf {Z}}}}_{\varvec{I,r}} \varvec{\in \tilde{\mathfrak g}}\) | Element \({\varvec{\exp \left( {\tilde{{\mathbf {Z}}}_{I,r}}\right) \in \tilde{G}}}\) | Subalgebra in \(\varvec{\tilde{\mathfrak g}}\) |
|---|---|---|
\(\tilde{\mathbf {Z}}_{1,r} := \begin{pmatrix} z_{1,r} &{} -z_{1,r} &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{pmatrix}\) | \(\exp \left( {\tilde{\mathbf {Z}}_{1,r}}\right) = \begin{pmatrix} \exp \left( {z_{1,r}}\right) &{} 1- \exp \left( {z_{1,r}}\right) &{} 0\\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \end{pmatrix}\) | \(\tilde{\mathfrak g}_1 = \langle \tilde{\mathbf {Z}}_{1,r} \rangle \), \(r \in \{1,2\}\) |
\(\tilde{\mathbf {Z}}_{2,r} := \begin{pmatrix} z_{2,r} &{} 0 &{} -z_{2,r} \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0\end{pmatrix}\) | \(\exp \left( {\tilde{\mathbf {Z}}_{2,r}}\right) = \begin{pmatrix} \exp \left( {z_{2,r}}\right) &{} 0 &{} 1-\exp \left( {z_{2,r}}\right) \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 \end{pmatrix}\) | \(\tilde{\mathfrak g}_2 = \langle \tilde{\mathbf {Z}}_{2,r} \rangle \), \(r \in \{1,2\}\) |