2 added a section on quantized universal enveloping algebras

First say we're taking about $*$-Hopf algebras, to start with.

Then, in any case, the answer is no. Any compact quantum group admits a biinvariant Haar measure. On a general Hopf algebra not even the existence of a left integral is granted (Hopf algebra with a left integral, if I remember correctly, are called coFrobenius). Even if left and right integral exists it is not granted that they coincide (again I think it is quite easy to find finite-dimensional examples of this: see van Daele The Haar measure on finite quantum groups, Proc. Amer. Math. Soc. 125, 3489-3500 (1997)).

One has certainly to enlarge the theory to locally compact quantum groups; I'm not completely sure whether one can obtain existence in such a case. I strongly doubt about unicity.

ADDED Drinfel'd-Jimbo quantization, by which I mean quantized universal enveloping algebras, are examples of non commutative Hopf * algebras that do not fall in the CQG setting; in fact they have a real group-like element $K$ and and easy computation shows that any left invariant integral $\phi$ should satisfy $\phi(K^2)=\phi(KK^*)=0$, thus cannot be positive and therefore there is no Haar measure. This should come as no surprise. The quantum duality principle allows to interpretate quantized universal enveloping algebras as quantizationof dual Poisson-Lie groups. Dual Poisson Lie groups of (standard Poisson-Lie) compact groups are not compact. This already shows that the setting of compact quantum groups is too narrow for the whole quantum group thing. If I remember correctly quantized universal enveloping algebras are algebraic quantum groups in the sense of Kustermans - van Daele, i.e. they have a Haar measure on the associated multiplier Hopf * algebra (which is non unital), in perfect agreement with the fact that they quantize non compact unimodular Lie groups.

1

First say we're taking about $*$-Hopf algebras, to start with.

Then, in any case, the answer is no. Any compact quantum group admits a biinvariant Haar measure. On a general Hopf algebra not even the existence of a left integral is granted (Hopf algebra with a left integral, if I remember correctly, are called coFrobenius). Even if left and right integral exists it is not granted that they coincide (again I think it is quite easy to find finite-dimensional examples of this: see van Daele The Haar measure on finite quantum groups, Proc. Amer. Math. Soc. 125, 3489-3500 (1997)).

One has certainly to enlarge the theory to locally compact quantum groups; I'm not completely sure whether one can obtain existence in such a case. I strongly doubt about unicity.