TITLE: Compact Quantum Groups from Hopf Algebras
QUESTION [7 upvotes]: For a compact quantum group $C_q[G]$, it was shown by Woronowicz that $C_q[G]$ contains a dense Hopf algebra generalising the algebra of representations of $G$. I am interested in the other way around, ie given a Hopf algebra $H$ (say a Drinfeld--Jimbo algebra if it makes things easier) can it always be completed to give a compact quantum group? If so, is this completion unique in anyway, or are there many ways to get a cqg from a Hopf algebra?

REPLY [7 votes]: A little side remark, concerning the last part of your question: Marc Rosso describes in
Marc Ross, Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel non commutatif, Duke Math. J. Volume 61, Number 1 (1990), 11-40. 
another way to get compact quantum groups from Hopf algebras. He defines inner products on the representation spaces of the deformed enveloping algebras of simple Lie groups (Drinfeld and Jimbo's $U_q(g)$'s), such that they form a concrete monoidal $W^*$ category. Applying Woronowicz' Tannaka-Krein duality he can recover compact quantum groups from this: $C(G_q)$, a deformation of the algebra of continuous functions on the simple Lie group $G$. The Hopf algebras $U_q(g)$ do not sit inside $C(G_q)$, but their restricted duals can be identified with the dense Hopf *-algebra contained in $C(G_q)$.