# 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. Jul 27 '18 at 3:43

I would maybe recommend some classical literature on quantum groups. I have in particular two books in mind:

Christian Kassel: Quantum groups

Anatoli Klimyk, Konrad Schmüdgen: Quantum Groups and Their Representations

Kassel brings quite detailed overview over both the Lie group $$SL(2)$$ as well as the Lie algebra $${\frak sl}(2)$$ and then describes in detail the duality between $$U({\frak sl}(2))$$ and $$SL(2)$$ in section V.7.

Klimyk–Schmüdgen describes this duality for a general Lie group $$G$$ and its Lie algebra $$\frak g$$ (although quite briefly) in Sections 1.2.5–1.2.6.

In order to bring yet another source, it is maybe worth reading also the book of Timmermann, where he describes duality of Hopf algebras in Section 1.4 and in particular discusses Lie groups and Lie algebras in Example 1.4.7.

Now as for the quantum group case, both Kassel and Klimyk–Schmüdgen study in detail the quantum groups $$SL_q(2)$$ and $$U_q({\frak sl}(2))$$ and then describe their duality (Kassel in §§VII.4–VII.5, Kl–Sch in §4.4).

Applications of Tannaka–Krein duality for quantum groups is the main focus of my research, but to be honest, I am not sure how is this related to this kind of duality. (But maybe it is. Could you maybe provide a link for the wiki article?)

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.