It is shown in [EO], Proposition 2.17, that a finite dimensional quasi-Hopf algebra with radical of codimension 1 is semisimple and 1-dimensional. On the other hand, there exist quasi-Hopf (in fact, Hopf) algebras, whose radical has codimension 2. Namely, it is known [N] that these are exactly the Nichols Hopf algebras H2n of
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Etingof et al. (2004) studied this question.
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