# Integrable modules of quantum groups.

Let $\widehat{\mathfrak{g}}$ be an affine Lie algebra and $U_q(\widehat{\mathfrak{g}})$ the corresponding quantum affine algebra. Let $V$ be an integrable module of $U_q(\widehat{\mathfrak{g}})$. That is, $V$ is a direct sum of its weight spaces and $E_i, F_i$ actions on $V$ are locally nilpotent. Can $V$ be written as a direct sum of irreducible $U_q(\widehat{\mathfrak{g}})$-modules? Thank you very much.

• When $q$ is generic, the category of integrable modules is semi-simple. This is not true when $q$ is a root of unity. – David Hill Mar 29 '18 at 15:23