Every 2-dimensional braided Hopf algebra over a quasi-Hopf algebra H is either the trivial k[C2] or a member of the explicitly parametrized family B_{σ,v}.
Bulacu, A structure theorem for quasi-Hopf bimodule coalgebras , Theory and Applications of Categories (TAC) 32(1) (2017), 1–30
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Biproduct Quasi-Hopf Algebras of Rank 2
Every 2-dimensional braided Hopf algebra over a quasi-Hopf algebra H is either the trivial k[C2] or a member of the explicitly parametrized family B_{σ,v}.