This is a follow up question to the insightful answer by Theo Johnson-Freyd to the question 105221.

This answer explained that the quantised enveloping algebra, $U_q(\mathfrak{g})$, (defined by a presentation due to Jimbo) could not be defined without a choice, say, of a Cartan subalgebra of $\mathfrak{g}$. The answer also made a comment to the effect that the category of (finite dimensional) representations of $U_q(\mathfrak{g})$ is a deformation quantisation of the category of (finite dimensional) representations of $\mathfrak{g}$. My question is whether this comment can be made precise?

The idea (as I understand it) is that we should put $q=\exp(h)$ and expand in power series over $h$. Then taking the degree zero part in $h$ gives the representation of $\mathfrak{g}$ and the linear part in $h$ gives some additional structure. If we think of a K-linear category as a K-algebra then this additional structure would be a Poisson structure. This would also be a replacement for Drinfeld's Poisson-Lie groups and Lie bialgebras. Then there would be further structure corresponding to the quasi-triangular structure.

Another line of thought is that Chern-Simons theory is canonically associated to $\mathfrak{g}$. Does this lead to a canonical construction of the representation theory of $U_q(\mathfrak{g})$? This would appear to work only for $q$ a root of unity.

My motivation for this question is that I would like to apply this to Vogel's universal Lie algebra where the Lie algebra is an object in a category and so any choice required for the construction of the quantised enveloping algebra would seem to be unavailable.

This is a follow up question to the insightful answer by Theo Johnson-Freyd to the question 105221.

This answer explained that the quantised enveloping algebra, $U_q(\mathfrak{g})$, (defined by a presentation due to Jimbo) could not be defined without a choice, say, of a Cartan subalgebra of $\mathfrak{g}$. The answer also made a comment to the effect that the category of (finite dimensional) representations of $U_q(\mathfrak{g})$ is a deformation quantisation of the category of (finite dimensional) representations of $\mathfrak{g}$. My question is whether this comment can be made precise?

The idea (as I understand it) is that we should put $q=\exp(h)$ and expand in power series over $h$. Then taking the degree zero part in $h$ gives the representation of $\mathfrak{g}$ and the linear part in $h$ gives some additional structure. If we think of a K-linear category as a K-algebra then this additional structure would be a Poisson structure. This would also be a replacement for Drinfeld's Poisson-Lie groups and Lie bialgebras. Then there would be further structure corresponding to the quasi-triangular structure.

Another line of thought is that Chern-Simons theory is canonically associated to $\mathfrak{g}$. Does this lead to a canonical construction of the representation theory of $U_q(\mathfrak{g})$? This would appear to work only for $q$ a root of unity.

My motivation for this question is that I would like to apply this to Vogel's universal Lie algebra where the Lie algebra is an object in a category and so any choice required for the construction of the quantised enveloping algebra would seem to be unavailable.