On Hopf algebras over the unique 12-dimensional Hopf algebra without the dual Chevalley property

Abstract

Let $\mathbb{k}$ be an algebraically closed field of characteristic zero. We determine all finite-dimensional Hopf algebras over $\mathbb{k}$ whose Hopf coradical is isomorphic to the unique 12-dimensional Hopf algebra $\mathcal{C}$ without the dual Chevalley property, such that the diagrams are strictly graded and the corresponding infinitesimal braidings are indecomposable objects in ${}_{\mathcal{C}}^{\mathcal{C}}\mathcal{Y}\mathcal{D}$. In particular, we obtain new Nichols algebras of dimension 18 and 36 and two families of new Hopf algebras of dimension 216.

Keywords:

• Generalized lifting method
• Hopf algebra
• Nichols algebra

2010 Mathematics Subject Classification:

• 16T05

Acknowledgements

The author is indebted to his supervisors Profs. Giovanna Carnovale and Naihong Hu so much for the kind help and continued encouragement. The author is grateful to Prof. G.-A. Garcia for helpful conversations and comments and also thanks Profs. G.-A. Garcia and L. Vendramin for providing the computation with GAP of Nichols algebras. The author would like to thank the referee for careful reading and helpful suggestions that largely improved the exposition.

Additional information

Funding

This paper was written during the visit of the author to University of Padova supported by China Scholarship Council10.13039/501100004543 (Grant No. 201706140160) and the NSFC10.13039/501100001809 (Grant No. 11771142).