Theory and Modern Applications
Table 2 The commutation relations of infinitesimal generators \(\mathcal{X}_{1}\), \(\mathcal{X}_{2}\), \(\mathcal{X}_{3}\)
\([\mathcal{X}_{i},\mathcal{X}_{j}]\) | \(\mathcal{X}_{1}\) | \(\mathcal{X}_{2}\) | \(\mathcal{X}_{3}\) | \(\mathcal{X}_{4}\) |
|---|---|---|---|---|
\(\mathcal{X}_{1}\) | \(\mathcal{X}_{1}\) | \(\mathcal{X}_{2}\) | \(\mathcal{X}_{3}\) | \(\epsilon \mathcal{X}_{1}+\mathcal{X}_{4}\) |
\(\mathcal{X}_{2}\) | \(\mathcal{X}_{1}\) | \(\mathcal{X}_{2}\) | \(\mathcal{X}_{3}\) | \(-\epsilon \mathcal{X}_{2}+\mathcal{X}_{4}\) |
\(\mathcal{X}_{3}\) | \(\mathcal{X}_{1}\) | \(\mathcal{X}_{2}\) | \(\mathcal{X}_{3}\) | \(3\epsilon \mathcal{X}_{3}+\mathcal{X}_{4}\) |
\(\mathcal{X}_{4}\) | \(e^{-\epsilon }\mathcal{X}_{1}\) | \(e^{\epsilon }\mathcal{X}_{2}\) | \(e^{-3\epsilon }\mathcal{X}_{3}\) | \(\mathcal{X}_{4}\) |