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Invertible elements of the Hopf algebra quantum $SU(2)$

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

(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?

asked Feb 22, 2021 at 22:09