Let k be an algebraically closed field of characteristic zero. We determine all finite-dimensional Hopf algebras over k whose Hopf coradical is isomorphic to the unique 12dimensional Hopf algebra C without the dual Chevalley property, such that the diagrams are strictly graded and the corresponding infinitesimal braidings are indecomposable objects in C C YD. In particular, we obtain new Nichols algebras of dimension 18 and 36 and two families of new Hopf algebras of dimension 216. , and as an algebra is generated by the elements a, b, g, x satisfying the relations in C cop , the relations in A op cop 1 and ag = ga, ax + ξ −2 xa = λ −1 θξ −2 (ba 3 − gb), bg = −gb, bx + ξ −2 xb = θξ −2 (a 4 − ga). The projective class ring and representation type of the Drinfeld double DWe study the representation type of D and the projective class ring of D, which is a subring of the Green ring. We refer to [ARS95] for the representation theory.

Let k be an algebraically closed field of characteristic zero. We construct several families of finite-dimensional Hopf algebras over k without the dual Chevalley property via the generalized lifting method. In particular, we obtain 14 families of new Hopf algebras of dimension 128 with non-pointed duals which cover the eight families obtained in our unpublished version, arXiv:1701.01991 [math.QA].

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In this paper, we determine the cocycle deformations and Galois objects for non-commutative and non-cocommutative Hopf algebras of dimension 16. We show that these Hopf algebras are pairwise twist inequivalent by calculating the higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of these Hopf algebras admit non-trivial cocycle deformations, though all of them have more than one Galois objects.