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I need prove that any element of $U(\mathfrak{sl}_2)$ can be represented by linear combination of elements $e^i h^j C^k$, where $C=ef+fe+\dfrac{h^2}{2}$.

$e=\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \ \ h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \ \ f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$.

First I think about $U(\mathfrak{sl}_2)$. Using PBW-theorem I know that the elements $e^ih^jf^k$ form a basis of $U(\mathfrak{sl}_2)$. But then $U(\mathfrak{sl}_2)$ is finite dimensional? I have $e^2=0,f^2=0, h^2=1$.

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Alex-omsk
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    Note that the relations $e^2 = 0$, $f^2 = 0$, $h^2 = 1$ hold in the associative algebra of $2\times 2$ matrices (over the respective ground field), which corresponds to the fundamental representation of $\mathfrak{sl}_2$, not in the universal enveloping algebra. The relations that hold in $U(\mathfrak{sl}_2)$ are those that can be derived solely from the bracket relations $[h,e]=2e$, $[h,f]=-2f$, $[e,f]=h$. – ivanpenev Jun 13 '14 at 23:01
  • Yes, I see it. But I don't know how to solve problem. – Alex-omsk Jun 15 '14 at 15:20

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