In first chapter of Lie Groups for Pedestrians by Lipkin, a method of generalization of irreducible tensor operators (and other features of the quantum mechanical angular momentum algebra) is given.

The statement is that as long as one can find a finite number of operators $X_\rho$ satisfying analogous commutation relations to those of the angular momentum operators in quantum mechanics, i.e.

$$[X_\rho,\;X_\sigma]=C^\tau_{\rho\sigma}X_\tau,$$

one can always find irreducible tensor operators, in analogy to those of angular momentum. One can then, in analogy to $J_z$, choose one (or several) operators to be diagonal in the desired representation. Furthermore, one can extract the analogy of the ladder operators $J_x\pm iJ_y$.

For angular momentum ($SO(3)$), irreducible tensor operators are given in terms of the relation

$$[J_z,T_{kq}]=qT_{kq},$$

where $q$ is the number of components and $k$ is the rank of the tensor. There are $2k+1$ values for $q$, which ranges from $-k$ to $k$.

Analogous tensor operators can be constructed starting from any algebra of the above form. Note that the crucial object is the Lie algebra, not the Lie group, which can be formulated as the group of continuous transformations given by

$$\psi^\prime=(1+i\epsilon X_\rho)\psi.$$

This is not a rigorous answer since I haven't worked out the proof myself. I can only recommend you to read the book.

In first chapter of Lie Groups for Pedestrians by Lipkin, a method of generalization of irreducible tensor operators (and other features of the quantum mechanical angular momentum algebra) is given.

The statement is that as long as one can find a finite number of operators $X_\rho$ satisfying analogous commutation relations to those of the angular momentum operators in quantum mechanics, i.e.

$$[X_\rho,\;X_\sigma]=C^\tau_{\rho\sigma}X_\tau,$$

it is always possible to find irreducible tensor operators. One can then, in analogy to $J_z$, choose one (or several) operators to be diagonal in the desired representation. Furthermore, one can extract the analogy of the ladder operators $J_x\pm iJ_y$.

For angular momentum ($SO(3)$), irreducible tensor operators are given in terms of the relation

$$[J_z,T_{kq}]=qT_{kq},$$

where $q$ is the number of components and $k$ is the rank of the tensor. There are $2k+1$ values for $q$, which ranges from $-k$ to $k$.

Analogous tensor operators can be constructed starting from any algebra of the above form. Note that the crucial object is the Lie algebra, not the Lie group, which can be formulated as the group of continuous transformations given by

$$\psi^\prime=(1+i\epsilon X_\rho)\psi.$$

This is not a rigorous answer since I haven't worked out the proof myself. I can only recommend you to read the book.

In first chapter of Lie Groups for Pedestrians by Lipkin, a method of generalization of irreducible tensor operators (and other features of the quantum mechanical angular momentum algebra) is given.

The statement is that as long as one can find a finite number of operators $X_\rho$ satisfying analogous commutation relations to those of the angular momentum operators in quantum mechanics, i.e.

$$[X_\rho,\;X_\sigma]=C^\tau_{\rho\sigma}X_\tau,$$

one can always find irreducible tensor operators, in analogy to those of angular momentum. One can then, in analogy to $J_z$, choose one (or several) operators to be diagonal in the desired representation. Furthermore, one can extract the analogy to ladder operators.

For angular momentum ($SO(3)$), irreducible tensor operators are given in terms of the relation

$$[J_z,T_{kq}]=qT_{kq},$$

where $q$ is the number of components and $k$ is the rank of the tensor. There are $2k+1$ values for $q$, which ranges from $-k$ to $k$.

Analogous tensor operators can be constructed starting from any algebra of the above form. Note that the crucial object is the Lie algebra, not the Lie group, which can be formulated as the group of continuous transformations given by

$$\psi^\prime=(1+i\epsilon X_\rho)\psi.$$

This is not a rigorous answer since I haven't worked out the proof myself. I can only recommend you to read the book.

In first chapter of Lie Groups for Pedestrians by Lipkin, a method of generalization of irreducible tensor operators (and other features of the quantum mechanical angular momentum algebra) is given.

The statement is that as long as one can find a finite number of operators $X_\rho$ satisfying analogous commutation relations to those of the angular momentum operators in quantum mechanics, i.e.

$$[X_\rho,\;X_\sigma]=C^\tau_{\rho\sigma}X_\tau,$$

one can always find irreducible tensor operators, in analogy to those of angular momentum. One can then, in analogy to $J_z$, choose one (or several) operators to be diagonal in the desired representation. Furthermore, one can extract the analogy of the ladder operators $J_x\pm iJ_y$.

For angular momentum ($SO(3)$), irreducible tensor operators are given in terms of the relation

$$[J_z,T_{kq}]=qT_{kq},$$

where $q$ is the number of components and $k$ is the rank of the tensor. There are $2k+1$ values for $q$, which ranges from $-k$ to $k$.

Analogous tensor operators can be constructed starting from any algebra of the above form. Note that the crucial object is the Lie algebra, not the Lie group, which can be formulated as the group of continuous transformations given by

$$\psi^\prime=(1+i\epsilon X_\rho)\psi.$$

This is not a rigorous answer since I haven't worked out the proof myself. I can only recommend you to read the book.