Theory and Modern Applications

# TableÂ 1 Commutator table

$$[ \boldsymbol{V}_{\boldsymbol{1}}, \boldsymbol{V}_{\boldsymbol{2}}]$$

$$\boldsymbol{X}_{\boldsymbol{1}}$$

$$\boldsymbol{X}_{\boldsymbol{2}}$$

$$\boldsymbol{X}_{\boldsymbol{3}}$$

$$\boldsymbol{X}_{\boldsymbol{4}}$$

$$\boldsymbol{X}_{\boldsymbol{1}}$$

0

$$f_{3} ' \frac{\partial }{\partial z} + f_{3}^{\prime \prime } \frac{\partial }{\partial u} + a_{1} \frac{\partial }{\partial p}$$

$$\frac{\partial }{\partial t} + a_{2} \frac{\partial }{\partial z} + a_{3} \frac{\partial }{\partial u} + a_{4} \frac{\partial }{\partial p}$$

$$a_{5} \frac{\partial }{\partial z} + a_{6} \frac{\partial }{\partial u} + a_{7} \frac{\partial }{\partial p}$$

$$\boldsymbol{X}_{\boldsymbol{2}}$$

$$-( f_{3} ' \frac{\partial }{\partial z} + f_{3}^{\prime \prime } \frac{\partial }{\partial u} + a_{1} \frac{\partial }{\partial p} )$$

0

$$-t f_{3} ' \frac{\partial }{\partial z} + a_{8} \frac{\partial }{\partial u} - \frac{2}{r^{2} v} \frac{\partial }{\partial v} + a_{9} \frac{\partial }{\partial p}$$

$$f_{3} \frac{\partial }{\partial z} + f_{3} ' \frac{\partial }{\partial u} + \frac{4}{r^{2} v} \frac{\partial }{\partial v} + a_{10} \frac{\partial }{\partial p}$$

$$\boldsymbol{X}_{\boldsymbol{3}}$$

$$-( \frac{\partial }{\partial t} + a_{2} \frac{\partial }{\partial z} + a_{3} \frac{\partial }{\partial u} + a_{4} \frac{\partial }{\partial p} )$$

$$-(-t f_{3} ' \frac{\partial }{\partial z} + a_{8} \frac{\partial }{\partial u} - \frac{2}{r^{2} v} \frac{\partial }{\partial v} + a_{9} \frac{\partial }{\partial p} )$$

0

$$a_{11} \frac{\partial }{\partial z} + a_{12} \frac{\partial }{\partial u} + a_{13} \frac{\partial }{\partial p}$$

$$\boldsymbol{X}_{\boldsymbol{4}}$$

$$-( a_{5} \frac{\partial }{\partial z} + a_{6} \frac{\partial }{\partial u} + a_{7} \frac{\partial }{\partial p} )$$

$$-( f_{3} \frac{\partial }{\partial z} + f_{3} ' \frac{\partial }{\partial u} + \frac{4}{r^{2} v} \frac{\partial }{\partial v} + a_{10} \frac{\partial }{\partial p} )$$

$$-( a_{11} \frac{\partial }{\partial z} + a_{12} \frac{\partial }{\partial u} + a_{13} \frac{\partial }{\partial p} )$$

0