Injectivity of the homomorphism θ: SU_q(2) → B

Prove that the *- and coalgebra homomorphism θ from Woronowicz’s Hopf *-algebra of the compact quantum group SU_q(2) to the dual Hopf algebra B of the discrete quantum group constructed in the paper is injective, thereby establishing that θ is a *-isomorphism and B is (Hopf *-algebra) isomorphic to SU_q(2).

Background

In Section 6, the authors define elements u_{ij} in the dual B and construct a *-homomorphism θ from the Hopf *-algebra of SU_q(2) to B by mapping the generators (α, γ) to (u_{11}, u_{21}). Proposition 5.6d shows that θ is surjective.

The identification of B with SU_q(2) hinges on the injectivity of θ. While surjectivity is established, the injectivity—and hence the full isomorphism—remains to be proven within the paper.