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  1. Abstract We classify all non-affine Hopf algebras H over an algebraically closed field k of characteristic zero that are integral domains of Gelfand–Kirillov dimension two and satisfy the condition Ext 1 H ( k , k ) ≠ 0. The affine ones were classified by the authors in 2010 (Goodearl and Zhang, J. Algebra 324 (2010), 3131–3168).