TITLE: Quantized Enveloping Algebras at $q=1$
QUESTION [10 upvotes]: As is well-known, the quantized enveloping algebra $U_q(\frak{sl}_2)$ is not well-defined when $q=1$ because of the relation
$$
[E,F] = \frac{K-K^{-1}}{q-q^{-1}}.
$$
To address this problem, one has an alternative formulation of $U_q(\frak{sl}_2)$: One removes the offensive relation, introduces an extra generator $G$, along with extra relations, 
$$
[G,E] = E(qK+q^{-1} K^{-1}), ~~~~~~~~~ [G,F] = -(qK + q^{-1} K^{-1})F,
$$
and 
$$
[E,F] = G, ~~~~~~~~ (q-q^{-1})G = K - K^{-1}.
$$
and canonical extensions of the Hopf algebra maps. This new algebra can be shown isomorphic to $U_q(\frak{sl}_2)$, yet is well-defined at $q=1$. It is not equal to $U(\frak{sl}_2)$, but to a double cover of $U(\frak{sl}_2)$.
What happens in the general $U_q(\frak{g})$ case? Again $U_q(\frak{g})$ is not well-defined since we the relation
$$
[E_i,F_j] = \delta_{ij}\frac{K_i-K^{-1}}{q-q^{-1}}.
$$
Judging by the representation theory, there should exist a reformulation of $U_q(\frak{g})$, where the offensive relations are removed, $n$ new generators $G_i$ (where $n$ is the rank of $\frak{g}$) are introduced, along with an extra set of new relations, which I would guess as
$$
[G_i,E_j] = \delta_{ij}E_i(qK_i+q^{-1} K_i^{-1}), ~~~ [G_i,F_j] = -\delta_{ij}(qK_i + q^{-1} K_i^{-1})F_i,
$$
and
$$
[E_i,F_j] = \delta_{ij}G_i, ~~~~~~~ (q-q^{-1})G_i = K_i - K_i^{-1}.
$$
This should then be equal to $U_q[\frak{g}$, and is clearly well-defined when $q=1$. I would then guess that at $q=1$ we have an $n$-fold cover of $U(\frak{g})$, which one consider as justification for the $2^n$ representations types of $U_q(\frak{g})$. Am I OK in reasoning here?

REPLY [8 votes]: Even if strictly speaking it's not true, it's always usefull to remember that morally $$K_i=q_i^{H_i}:=\exp(H_i\log(q_i))$$ where $q_i=q^{d_i}$ for some integer $d_i$ attached to the Cartan matrix of $\mathfrak g$, and where $H_i$ is the $i$th Cartan generator of $\mathfrak g$. This explain why the classical limit of $K_i$ should be 1.
Now set 
$$G_i=\frac{K_i-K_i^{-1}}{q_i-q_i^{-1}}$$
Then using again this heuristic it's easliy seen that you have
$$\lim_{q\rightarrow 1} G_i= \lim_{q\rightarrow 1}\frac{q_i^{H_i}-q_i^{- H_i}}{q_i-q_i^{-1}}= H_i$$
It's now rather clear how to prove that it indeed work, since $G_i$ is precisely the R.H.S. of the relation you want to modify, so just replace it by
$$[E_i,F_j]=\delta_{i,j} G_i$$
and add the relaiton
$$(q-q^{-1})G_i=K_i-K_i^{-1}$$
which is nothing but the definition of $G_i$. That you get the same algebra is obvious because you did not really change anything. The former relation does not depend on $q$ anymore so gives you the appropriate relation at the classical limit, and the latter relation just become trivial.
Then following again the heuristic that $K_i$ should goes to 1 at the limit, take the quotient by the ideal generated by $K_i-1$, and you get the classical envelopping algebra.