Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
$\begingroup$When $q$ is a non-zero, non root of unity, the answer is yes: see Chapter I.8 in Brown and Goodearl, Lectures on Algebraic Quantum Groups. I'm not sure about the root of unity case.$\endgroup$