I have a general question regarding quantum groups. It seems to me that the representation theory of the algebra $\mathcal{U}_q(\widehat{\mathfrak{sl}}_{e-1})$ has many parallels with the representation theory of $\mathfrak{sl}_{e-1}$. Why is it this algebra which shares so much in common with $\mathfrak{sl}_{e-1}$ rather than its universal enveloping algebra $\mathcal{U}_q(\mathfrak{sl}_{e-1})$? ($\mathcal{U}_q(\widehat{\mathfrak{sl}}_{e-1})$ is the quantised universal enveloping algebra of the Lie algebra $\mathfrak{sl}_{e-1}\otimes \mathbb{C}[t,t^{-1}]\oplus \mathbf{C}c\oplus \mathbf{C}d$).

**Edit:** Let me try to formulate the question more precisely. Let $\mathfrak{g}=\mathfrak{sl}_{e-1}\otimes \mathbb{C}[t,t^{-1}]\oplus \mathbb{C}c\oplus \mathbb{C}d$ and let $U_q(\mathfrak{g})$ be its quantised universal enveloping algebra. Then the representation theory of $U_q(\mathfrak{g})$ has a lot in common with the representation theory of $\mathfrak{sl}_{e-1}$, for example there is a theory of integrable highest weight modules for the two objects and their finite-dimensional irreducible modules have a similar appearance. I am wondering why it is this algebra which shares properties with $\mathfrak{sl}_{e-1}$, rather than its own quantised universal enveloping algebra $U_q(\mathfrak{sl}_{e-1})$. If this doesn't make sense, then I mustn't understand something!