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?



Compact Quantum Groups from Hopf Algebras
I think this was solved in the paper:


MR1310296 (95m:16029)  

Dijkhuizen, Mathijs S.(NL-MATH); Koornwinder, Tom H.(NL-AMST-CS)  

CQG algebras: a direct algebraic approach to compact quantum groups. (English summary)  

Lett. Math. Phys. 32 (1994), no. 4, 315–330. 


They show that a Hopf $\*$-algebra $A$ is the maximal Hopf $\*$-algebra of a compact quantum group in the Woronowicz sense if (and only if) $A$ is spanned by the matrix coefficients of its finite-dimensional unitary (irreducible) corepresentations. The "only if" part was shown by Woronowicz, of course.


They also show that a Hopf $\*$-algebra has this property if and only if it admits a positive definite Haar functional (in the sense Andreas talks about).


You also ask about uniqueness-- this occurs if and only if $A$ is "coamenable" (see various papers by Bedos, Tuset and coauthors). A classical example comes from a discrete group $G$ with Hopf $\*$-algebra $\mathbb C[G]$ (under convolution product). Then there is the maximal $C^\*$ completion $C^\*(G)$ and the completion of $\mathbb C[G]$ acting on $\ell^2(G)$ by the left-regular representation, leading to $C^\*\_r(G)$. The quotient map $C^\*(G) \rightarrow C^\*\_r(G)$ is injective if and only if $G$ is amenable.


