# Decomposition of representations of the Virasoro algebra under $sl(2)$

The Virasoro algebra has a finite $sl(2)$ sub-algebra generated by $L_{-1}$, $L_0$ and $L_{+1}$. Let's consider a unitary highest weight representation of the Virasoro algebra with conformal weight $h>0$ and central charge $c>1$. How does this representation decompose under the $sl(2)$ sub-algebra?

It is clear that there is an invariant $sl(2)$ sub-module of weight $h$ consisting of the highest weight state $|h\rangle$ and the descendants $L_{-1}|h\rangle$, $L_{-1}L_{-1}|h\rangle$, $\dots$ However, there are many other descendants, such as $L_{-2}|h\rangle$, $L_{-3}|h\rangle$, etc. What $sl(2)$ modules will these states sit in? In particular, is the full $sl(2)$ representation completely reducible, so that it can be written as a sum of irreducible $sl(2)$ representations? Can anything general be said about the multiplicity of the $sl(2)$ representations appearing in the decomposition?

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Maybe this question belongs on math.SE, but there are hardly any questions about the Virasoro algebra at that site, while physics.SE has a whole bunch, so I think there is a bigger chance of getting an answer here. –  Olof Aug 7 at 13:19
Nah, I'd say the question should stay here, as these issues are of huge interest in theoretical physics, +1 BTW :-) –  Dilaton Aug 7 at 15:24
I did not understand a word of this reference, but the title seems to be related to your question... –  Trimok Aug 7 at 16:28
@Trimok: Thanks for the reference. I agree that it looks a bit dense but useful. –  Olof Aug 7 at 17:17