If $A$ is a bialgebra, Gerstenhaber and Schack in Bialgebra cohomology, deformations, and quantum groups defined a double complex $$C_{bi}^{p,q}\ =\ \text{Hom}_k\left(A^{\otimes p}, A^{\otimes q}\right), \hspace{15mm} p,q\ge 1$$ with $d$ given by the bar complex on $(A^{\otimes p})$ and the cobar complex on $(A^{\otimes q})$. First-order deformations of $A$ are classified by the second cohomology of this complex (after shifting it one down). This is plausible: such classes have representatives $$m\ \in\ \ \text{Hom}(A^{\otimes 2},A), \hspace{10mm} \Delta\ \in\ \text{Hom}_k(A,A^{\otimes 2}),$$ which will be first-order deformations to the product and coproduct on $A$.

It is well known that $A=U(\mathfrak{g})$ has a unique nontrivial one-parameter deformation: the quantum group $U_q(\mathfrak{g})$, if $\mathfrak{g}$ is a semisimple finite dimensional Lie algebra. So we get a first-order deformation.

Question: Where can I see a computation of $\text{H}^\bullet_{bi}(U(\mathfrak{g}))$, and what the generating class $[U_q(\mathfrak{g})]\in \text{H}^2_{bi}(U(\mathfrak{g}))\simeq \mathbf{C}$ is?

My question below is about how to view the quantum group $U_q(\mathfrak{g})$ as a bialgebra cohomology class.

Background: If $A$ is a bialgebra, Gerstenhaber and Schack in Bialgebra cohomology, deformations, and quantum groups defined a double complex $$C_{bi}^{p,q}\ =\ \text{Hom}_k\left(A^{\otimes p}, A^{\otimes q}\right), \hspace{15mm} p,q\ge 1$$ with $d$ given by the bar complex on $(A^{\otimes p})$ and the cobar complex on $(A^{\otimes q})$. First-order deformations of $A$ are classified by the second cohomology of this complex (after shifting it one down). This is plausible: such classes have representatives $$m\ \in\ \ \text{Hom}(A^{\otimes 2},A), \hspace{10mm} \Delta\ \in\ \text{Hom}_k(A,A^{\otimes 2}),$$ which will be first-order deformations to the product and coproduct on $A$.

Question: Where can I see a computation of $\text{H}^\bullet_{bi}(U(\mathfrak{g}))$, and what the generating class $$[U_q(\mathfrak{g})]\ \in\ \text{H}^2_{bi}(U(\mathfrak{g}))\ \simeq \ \mathbf{C}$$ is?