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

Abstract

Let $\mathds{k}$ be an algebraically closed field of characteristic zero. We determine all finite-dimensional Hopf algebras over $\mathds{k}$ 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 ${}_{\mathcal{C}}^{\mathcal{C}}\mathcal{YD}$. In particular, we obtain new Nichols algebras of dimension $18$ and $36$ and two families of new Hopf algebras of dimension $216$.

24 pages

References in corpus (6)

• Nichols algebras that are quantum planes
• Pointed and copointed Hopf algebras as cocycle deformations
• Pointed Hopf algebras as cocycle deformations
• Some Hopf algebras of dimension $72$ without the Chevalley property
• Finite-dimensional Hopf algebras over the smallest non-pointed basic Hopf algebra
• Eight classes of new Hopf algebras of dimension $128$ without the Chevalley property

Cited by in corpus (2)

• Finite-dimensional Hopf algebras over the smallest non-pointed basic Hopf algebra
• On $4n$-dimensional neither pointed nor semisimple Hopf algebras and the associated weak Hopf algebras