# 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.

• arxiv.org/abs/hep-th/9111043 This survey paper has an explanation of the fact that $C^\infty(G)$ embeds into $U(\mathfrak{g})^*$. When we quantize things, the group $G$ should become a non-commutative space (in the sense of non-commutative geometry), so the duality in this case should be understood more abstractly. – Henry Jul 27 '18 at 3:43