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## Summary:

- I need to understand this passage from "An Elementary Introduction to Groups and Representations" Brian C. Hall

## Main Question or Discussion Point

Hi all I need to understand the following passage from Hall link page 78 :

Some notation first:

Basis for ##sl(2;C)##:

##H=\begin{pmatrix} 1&0\\0&−1\end{pmatrix} ;X=\begin{pmatrix} 0&1\\0&0\end{pmatrix} ;Y=\begin{pmatrix} 0&0\\1&0\end{pmatrix} ##

which have the commutation relations

##[H,X] = 2X ~ ~, [H,Y] =−2Y ~ , [X,Y] =H ##

##π(X)## acts as the raising operator such that:

##π(H)π(X)u= (α+ 2)π(X)u ##

##π(Y)## acts as the lowering operator such that:

##π(H)π(Y)u= (α−2)π(Y)u ##

There is some N≥0 such that ##π(X)^Nu \neq 0##

but ##π(X)^{N+1}u= 0 ##

We define ##u_0=π(X)^Nu ## then

##(H)u_0=λu_0##

##π(X)u_0= 0 ##

Now, by definition

##u_{k+1}=π(Y)u_k##

Using ##π(H)u_k= (λ−2k)u_k## and induction we have

##π(X)u_{k+1}=π(X)π(Y)u_{k}

\\= (π(Y)π(X) +π(H))u_k

\\=π(Y) [kλ−k(k−1)]u_{k−1}+ (λ−2k)u_k

\\= [kλ−k(k−1) + (λ−2k)]u_k##

I don't understand how to get ##kλ−k(k−1)]## at the third passage and why ## (λ−2k)u_k## should be zero to get

Some notation first:

Basis for ##sl(2;C)##:

##H=\begin{pmatrix} 1&0\\0&−1\end{pmatrix} ;X=\begin{pmatrix} 0&1\\0&0\end{pmatrix} ;Y=\begin{pmatrix} 0&0\\1&0\end{pmatrix} ##

which have the commutation relations

##[H,X] = 2X ~ ~, [H,Y] =−2Y ~ , [X,Y] =H ##

##π(X)## acts as the raising operator such that:

##π(H)π(X)u= (α+ 2)π(X)u ##

##π(Y)## acts as the lowering operator such that:

##π(H)π(Y)u= (α−2)π(Y)u ##

There is some N≥0 such that ##π(X)^Nu \neq 0##

but ##π(X)^{N+1}u= 0 ##

We define ##u_0=π(X)^Nu ## then

##(H)u_0=λu_0##

##π(X)u_0= 0 ##

Now, by definition

##u_{k+1}=π(Y)u_k##

Using ##π(H)u_k= (λ−2k)u_k## and induction we have

##π(X)u_{k+1}=π(X)π(Y)u_{k}

\\= (π(Y)π(X) +π(H))u_k

\\=π(Y) [kλ−k(k−1)]u_{k−1}+ (λ−2k)u_k

\\= [kλ−k(k−1) + (λ−2k)]u_k##

I don't understand how to get ##kλ−k(k−1)]## at the third passage and why ## (λ−2k)u_k## should be zero to get

**##π(X)u_{k+1}= [kλ−k(k−1)]u_k##.**
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