Table 3 Elements of the Lie algebra \({\mathfrak t}\) and the associated Lie group T
Element \(\varvec{\mathbf {T}}_{\varvec{I,r}} \varvec{\in {\mathfrak t}}(3,\mathbb {C})\) | Element \(\varvec{\exp \left( {\mathbf {T}_{I,r}}\right) \in T}(3,\mathbb {C})\) | Subalgebra in \(\varvec{\mathfrak t}(3,\mathbb {C})\) |
|---|---|---|
\(\mathbf {T}_{1,r} := \begin{pmatrix} t_{1,r}\\ t_{1,r}\\ 0\end{pmatrix}\) | \(\exp \left( {\mathbf {T}_{1,r}}\right) = \begin{pmatrix} t_{1,r}\\ t_{1,r}\\ 0\end{pmatrix}\) | \(\mathfrak t_1 = \langle \mathbf {T}_{1,r} \rangle \), \(r \in \{1,2\}\) |
\(\mathbf {T}_{2,r} := \begin{pmatrix} t_{2,r}\\ 0\\ t_{2,r}\end{pmatrix}\) | \( \exp \left( {\mathbf {T}_{2,r}}\right) = \begin{pmatrix} t_{2,r}\\ 0\\ t_{2,r}\end{pmatrix}\) | \(\mathfrak t_2 = \langle \mathbf {T}_{2,r} \rangle \), \(r \in \{1,2\}\) |
\(\mathbf {T}_{3,r} := \begin{pmatrix} t_{3,r}\\ 0\\ 0\end{pmatrix}\) | \(\exp \left( {\mathbf {T}_{3,r}}\right) = \begin{pmatrix} t_{3,r}\\ 0\\ 0\end{pmatrix}\) | \(\mathfrak t_3 = \langle \mathbf {T}_{3,r} \rangle \), \(r \in \{1,2\}\) |