Let $SU_q(2)$ be the (polynomial) Hopf algebra introduced by Woronocicz called **the quantum special unitary group**. For details see

https://en.wikipedia.org/wiki/Compact_quantum_group

(Note that on the Wikipedia page it is the $C^*$-algebra that is discussed, but this question is about the dense Hopf algebra of the $C^*$-algebra.) 

Does $SU_q(2)$ contain any (non-unital) invertible elements?