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.

  • $\begingroup$ 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. $\endgroup$ – Henry Jul 27 '18 at 3:43

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.


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