For $\mathfrak{g}$ a semisimple Lie algebra, we may define its (untwisted) loop algebra as $L(\mathfrak{g}) = \mathfrak{g} \otimes \mathbb{C}\lbrack t,t^{-1} \rbrack$. Let $\mathcal{F}$ be the category of finite-dimensional $L(\mathfrak{g})$-modules.

It is stated throughout the literature (e.g. [1]) that $\mathcal{F}$ is not semisimple. While it is straightforward to find an object in $\mathcal{F}$ that is reducible but not decomposable, I would like to know whether a purely categorical/algebraic (i.e. abstract) proof of this fact is known?

[1] Senesi, P. 2009. Finite-dimensional Representation Theory of Loop Algebras: A Survey. (https://arxiv.org/abs/0906.0099).


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