Abstract
We study the Hopf *-algebra structures on the Hopf algebra H(1, q) over ℂ. It is shown that H(1, q) is a Hopf *-algebra if and only if |q| = 1 or q is a real number. Then the Hopf *-algebra structures on H(1, q) are classified up to the equivalence of Hopf *-algebra structures.
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