Let $\mathfrak{g} = \displaystyle\varinjlim_{n} \mathfrak{g}_n$ be a locally simple Lie algebra. Let $M$ be an integrable module over $\mathfrak{g}$, that is, for every $n$, as a $\mathfrak{g}_n$ - module ($M\downarrow \mathfrak{g}_n$), $M$ is a sum of simple finite dimensional $\mathfrak{g}_n$ - modules. Let $M_{n}^1, \ldots, M_n^{k_n}$ be the isotypical components of $M\downarrow\mathfrak{g}_n$.

I remember some time ago reading about some criterion for $M$ to be simple interms of the isotypical components $M_{n}^1, \ldots, M_n^{k_n}$. I can't seem to find that paper anymore. So my question is this: can someone point me to that paper or reproduce the criterion?

Thank you.


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