What are the v_j vectors in Cahn's representation of the Lie algebra of SO(3)?

[jdstokes]

Hi all, I asked this on the Quantum Physics board but didn't get a response.

I'm reading Cahn's book on semi-simple lie algebras and their representations.

http://www-physics.lbl.gov/~rncahn/book.html

In chapter 1, he attempts to build a (2j+1)-dimensional representation $T$ of the Lie algebra of SO(3) starting with the abstract commutation relations

$[T_z,T_+] = T_+, \quad [T_z,T_-] = - T_-,\quad [T_+,T_-] = 2T_z$ Eq (I.14).

He begins by defining the action of $T_z,T_+$ on the vector $v_j$ by

$T_z v_j = j v_j, \quad T_+ v_j = 0$

but he does not explain what the $v_j$'s are. How does one even know that such vectors exist?

Any help would be greatly appreciated.

 

[George Jones]

Hi all, I asked this on the Quantum Physics board but didn't get a response.

I'm reading Cahn's book on semi-simple lie algebras and their representations.

http://www-physics.lbl.gov/~rncahn/book.html

In chapter 1, he attempts to build a (2j+1)-dimensional representation $T$ of the Lie algebra of SO(3) starting with the abstract commutation relations

$[T_z,T_+] = T_+, \quad [T_z,T_-] = - T_-,\quad [T_+,T_-] = 2T_z$ Eq (I.14).

He begins by defining the action of $T_z,T_+$ on the vector $v_j$ by

$T_z v_j = j v_j, \quad T_+ v_j = 0$

but he does not explain what the $v_j$'s are. How does one even know that such vectors exist?

Any help would be greatly appreciated.

[EDIT]I've decided to answer on the quantum physics forum[/EDIT]