Let $a, b \in \mathbb{C}$ and let $V$ be an infinite dimensional vector space with basis $\{v_i | I \in \mathbb{Z}\}$. Define linear transformations $h$ and $f$ on $V$ by $$h(v_i) =(a+2i) v_i$$ and $$f(v_i) = v_{i-1} \forall i \in \mathbb{Z}$$.

Question 1: Set $e(v_0) = bv_1$. Show that there is a unique way to define $e(v_i)$. Show that there is a unique way to define $e(v_i)$ for all $i \in \mathbb{Z}$ in order to make V a module over $\mathcal{sl_2}$

Answer: I was able to figure it out by induction and I got $$e(v_i) = (b-i(a+i+1))v_{i+1}$$

Question 2: Show that any non-zero $sl_2$ sub-module $U$ of $V$ contains some $v_i$

Answer: I don't think I know what to do here but my understanding is to find a finite dimensional submodule $U$ of $V$ such that the linear transformation $h$ sends element in $U$ to some elements in $V$.

Question 3: Show that $V$ is reducible for if $$b= ja + j(j+1)$$ for some $j \in \mathbb{Z}$ and $V$ is irreducible if $b \neq ja + j(j+1)$ for all $j \in \mathbb{Z}$

Answer: if $b= ja + j(j+1)$ then $$e(v_j) =0$$ and showing that the

$$W = span \{.... \cdots....v_{j-1} , v_j\}$$ is a submodule of $V$ is enough. For the second part, I am guessing I have to show that V is generated by a single element say $v_{j+1}$ but I don't know how to show this.

P.S: I am pretty much new to Lie algebra. I will appreciate hints, suggestions, corrections and answers. Thanks.


You should state that $\text{sl}_2(\mathbb{C})$ is the $\mathbb{C}$-span of $h$, $e$, and $f$ with $[e,f]=h$, $[h,e]=2e$, and $[h,f]=-2f$. Not everybody can understand what $h$, $e$, and $f$ are.

More generally, what Question 2 asks you to do is to prove that, if $\mathfrak{h}$ is a Cartan subalgebra of a finite-dimensional semisimple Lie algebra $\mathfrak{g}$ and $M$ is an $\mathfrak{h}$-weight $\mathfrak{g}$-module, then any $\mathfrak{g}$-submodule of $M$ is an $\mathfrak{h}$-weight $\mathfrak{g}$-module. To prove the version in this question, suppose that $u$ is a nonzero element of a nonzero $\text{sl}_2(\mathbb{C})$-submodule $U$ of $V$. Then, $u=t_1\,v_{i_1}+\ldots+t_k\,v_{i_k}$ for some $t_1,\ldots,t_k\in\mathbb{C}\setminus\{0\}$ and for some integers $i_1<i_2<\ldots<i_k$. If $k=1$, we are done. Assume now that $k>1$.

First, observe that $$h^p\cdot u=\sum_{\mu=1}^k\,t_\mu\,\left(a+2i_\mu\right)^p\,v_{i_\mu}\,.$$ Take $P(X):=X-\left(a+2i_k\right)$ and $Q(X):=\prod_{\mu=1}^{k-1}\,\Big(X-\left(a+2i_\mu\right)\Big)$. Then, there exist polynomials $f(X),g(X)\in\mathbb{C}[X]$ such that $f(X)\,P(X)+g(X)\,Q(X)=1$. Now, we have $$g(h)\,Q(h)\cdot u=\sum_{\mu=1}^{k-1}\,t_i\,g(h)\,Q(h)\cdot v_{i_\mu}+t_k\,\big(1-f(h)\,P(h)\big)\cdot v_{i_k}\,.$$ Show that, for $\mu=1,2,\ldots,k-1$, $Q(h)\cdot v_{i_\mu}=0$ and $P(h)\cdot v_{i_k}=0$. Consequently, $t_k\,v_{i_k}=g(h)\,Q(h)\cdot u\in U$, whence $v_{i_k}\in U$. (To be rigorous, $h^p$, $f(h)$, $g(h)$, $P(h)$, and $Q(h)$ are viewed as elements of the universal enveloping algebra $\mathfrak{U}\big(\text{sl}_2(\mathbb{C})\big)$.)

For Question 3, use Question 2. Show that, if $b\neq j(a+j+1)$ for any $j\in\mathbb{Z}$, then $e^p\cdot v_i$ is a nonzero scalar multiple of $v_{i+p}$ for any $p$ and $i$.

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  • $\begingroup$ Do you happen to know if the "preservation of Jordan normal form" theorem in Fulton and Harris holds for infinite dimensional representations? (If it does, then for question (2), $H$ would be diagonalisable when acting on $U$, so surely $U$ will have a basis of $v_i$'s?) $\endgroup$ – Kenny Wong Mar 7 '17 at 18:28
  • $\begingroup$ Can you state that theorem? I don't have access to the book at the moment. $\endgroup$ – Batominovski Mar 7 '17 at 18:31
  • $\begingroup$ Thank you Batominovski. I should be able to figure it out now. $\endgroup$ – Jaynot Mar 7 '17 at 18:37
  • $\begingroup$ "If $\rho : \mathfrak{g} \to \mathfrak{gl}(V)$ is any representation of a semisimple Lie algebra $\mathfrak{g}$, then $\rho(X_s)$ is the semisimple part of $\rho(X)$ and $\rho(X_n)$ is the nilpotent part of $\rho(X)$." (Here $X = X_s + X_n$ is the Jordan decomposition of $X$, viewed as a matrix acting on $\mathfrak{g}$ itself via the adjoint rep. Apparently $X_s$ and $X_n$ are themselves in $\mathfrak{g}$.) $\endgroup$ – Kenny Wong Mar 7 '17 at 18:37
  • $\begingroup$ It's not clear the representation $V$ in this theorem is assumed to be finite. $\endgroup$ – Kenny Wong Mar 7 '17 at 18:38

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