# Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional modules of $U_q(g)$?

Let $g$ be a complex simple Lie algebra and $U_q(g)$ the corresponding quantum group.

Is the category of all finite dimensional modules of $g$ equivalent to the category of all finite dimensional modules of $U_q(g)$?

It seems that when $q$ is a root of unity, the category of all finite dimensional modules of $g$ is equivalent to the category of all finite dimensional modules of $U_q(g)$. When $q$ is not a root of unity, the category of all finite dimensional modules of $g$ is not equivalent to the category of all finite dimensional modules of $U_q(g)$. Is this true? Thank you very much.

• I believe you have it backward. For generic $q$, the category of finite dimensional $\mathfrak{g}$ modules is equivalent to that of $U_q(\mathfrak{g})$ (which is why crystals are useful). When $q$ is a root of unity, the center of $U_q(\mathfrak{g})$ becomes larger than for $U(\mathfrak{g})$ and the finite dimensional representations are different. – David Hill Nov 10 '15 at 16:20
• @David Hill, yes, thank you very much. – LJR Nov 11 '15 at 2:21