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Let $\frak{g}$ be a complex semi-simple Lie algebra of rank $n$, and $U_q(\frak{g})$ the corresponding Drinfeld-Jimbo algebra. As is well-known, for $q$ not a root of unity, the irreducible finite-dimensional representations of $U_q(\frak{g})$ are divided into $2^n$ types, corresponding to elements of the $n^{\text{th}}$-Cartesian power of the set {$+,-$}. The classical representations are recovered as those corresponding to $(1,1, \dots, 1)$, and are called Type 1. One defines the coordinate algebra ${\cal O}_q[G]$ to be the direct sum of the coordinate functions of the Type 1 irreducible representation. When $q = 1$, the algebraic Peter-Weyl tells us that we recover the coordinate algebra of the connected compact semi-simple Lie group $G$ corresponding to $\frak{g}$.

With the background recalled, let me now enquire:

(i) What happens when one takes the direct sum of the coordinate algebras all the irreducible representations, ie irreps of all types? One should recover the Hopf dual of $U_q(\frak{g})$. Thus, I would guess that this means that ${\cal O}_q[G]$ is strictly smaller that the Hopf dual?

(ii) What happens when $q$ is a root of unity? The irrep theory becomes a lot more complicated, so it is not clear that one can still define ${\cal O}_q[G]$ as one does for roots of unity. However, the construction of ${\cal O}_q[G]$ via $R$-matrices still makes sense (if I have understood correctly). So can one still define ${\cal O}_q[G]$ as some subalgebra of the Hopf dual?

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Regarding (ii), I believe the answer is basically yes and you might consult Chapter III.7 of the Brown-Goodearl book "Lectures on Algebraic Quantum Groups." – Casteels Apr 10 '13 at 21:10
@Janos: Your questions cover a lot of ground, but there is also a lot of relevant literature over 25+ years. It's not clear what you have looked at, but for instance there are many papers, surveys, books: see Chapters 5 and 7 of Jantzen's 1996 book Lectures on Quantum Groups, along with fundamental papers by DeConcini-Kac-Procesi, Lusztig, and the book by Chari-Pressley, etc. In the root of unity case, the function akgebra and enveloping algebra q-analogues certainly diverge sharply as their representation theory shows. But I can't offer a full overview. – Jim Humphreys Apr 10 '13 at 22:20
P.S. Add a tag rt.representation-theory? – Jim Humphreys Apr 10 '13 at 22:21
I am a bit confused by the fact that you refer to "connected compact semisimple Lie group " when $\mathfrak g$ is a complex semisimple Lie algebra. Shouldn't algebraic Peter-Weyl refer to $G$ be the unique connected, simply connected, complex algebraic group? – Nicola Ciccoli Apr 12 '13 at 14:46
One reason why I asked the previous is the fact that it is not completely clear to me what you are comparing, since for any ficed $\mathfrak g$ you have many $G$'s, all such that the corresponding ${\cal O}(G)$ sits inside the Hopf dual $U(\mathfrak g)^0$, therefore certainly not for any group $G$ such that $Lie(G)=\mathfrak g$ you have that ${\cal O}(G)$ is equal to the Hopf dual, already at the classical case. – Nicola Ciccoli Apr 12 '13 at 18:09

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