Explicit correspondence between classical double and quantum double

Question

Proposition 12.3 of Etingof and Schiffmann's "Lectures on Quantum Groups" states the following claim.

Proposition 12.3. Let $H$ be a quantized enveloping algebra and let $\mathfrak{g}$ be the quasi-classical limit of $H$. Then $D(H)$ is a quantization of $D\mathfrak{g}$.

Is there a specific way to write down the construction of this proposition when $\mathfrak{g}$ is a borel subalgebra of $\mathfrak{sl}(2, \mathbb{C})$?

I found a quantum double construction of $U_h(\mathfrak{b}_+)$ in Proposition 14 on p.258 of Klimyk and Schmiidgen's book "Quantum Groups and Their Representations".

The proposition gives the following commutation relation.

$$[H,\tilde{H}]=0, \quad[H,\tilde{F}]=-2\tilde{F}$$ $$[E,\tilde{H}]=-2\hbar E, \quad[E,\tilde{F}]= \hbar\dfrac{e^{\hbar H}-e^{-\hbar \tilde{H}}} {e^{\hbar}-e^{-\hbar \tilde{H}}} $$

I think that if we set $\hbar$ in this proposition to 0, we do not get $D\mathfrak{b}_{+}$.

Answer 1

The thing is that in the infinite dimensional case there is no such thing as "the" double: for any Hopf algebra $K$, given a non-degenerate Hopf pairing on $H\times K$ you can define a double $D(H,K)$.

A natural choice in that case is to take $K$ to be the full dual of $H$: this is a so-called Quantum Formal Series Hopf Algebra (QFSHA), a quantization of the formal Poisson group associated with $\mathfrak g$. This gives I think the double of Klimyk-Schmiidgen.

Now by the so-called quantum duality principle (https://arxiv.org/abs/math/9909071) there is a functor $K\mapsto K^\vee$ from QFSHA's to QUEA's given by, roughly speaking, dividing the generators of $K$ by $\hbar$. $K^\vee$ is then a quantization of the Lie bialgebra $\mathfrak g^*$. The pairing should extend to $K^{\vee}$ and the double in that case should be a QUEA quantizing the double of $\mathfrak g$. This I think gives the algebra with the same relations as the one you gave but without those extra $\hbar$. This is spelled out in section 4.4 of https://arxiv.org/abs/q-alg/9506005.