Duality between universal enveloping algebra and algebras of functions There is a well known duality (of Hopf algebras)  between universal enveloping algebra $U(\mathfrak{g})$ of a complex Lie algebra  $\mathfrak{g}$ of a compact group $G$ and the algebra of continuous functions $C(G)$. 
My question is, is there in the literature any place where this is presented in some detail? Bonus: also some references pointing to the generalization of this fact to quantized universal enveloping algebras?
Wikipedia says it is related to Tannaka-Krein theory, but I don`t know much about it and from a preliminary search I found nothing about this duality in the texts.

I would suggest Hochschild's "Basic Theory of Algebraic Groups and Lie Algebras". It is about affine algebraic groups in general and not just compact Lie groups, but I think it is worthy to have a look there.
