Some classifications of finite-dimensional Hopf algebras over the Hopf algebra H_b:x²y of Kashina

2026/02/01 by Yihan Wu, Hengyi Wang, Naihong Hu
Mathematics · #math.QA

paper · pdf

published as Milan Journal of Mathematics (2026) · 30 pages

arxiv created 2026/02/01 · arxiv updated 2026/07/31

Abstract

Let H be the 16-dimensional nontrivial (namely, noncommutative and noncocommutative) semisimple Hopf algebra H_b:x²y classified by Kashina. We figure out all simple Yetter-Drinfeld H-modules, and then determine all finite-dimensional Nichols algebras satisfying the constraint condition B(V)≅ \bigotimes_{i∈ I}B(V_i), where V=\bigoplus_{i∈ I}V_i, each V_i is a simple object in _H^HYD. Finally, we describe some liftings of the corresponding Radford biproducts B(V)\sharp H, which provide some classifications of finite dimensional Hopf algebras with H as their coradical.

Citations

• Pointed Hopf algebras of odd dimension and Nichols algebras over solvable groups
• Finite-dimensional Hopf algebras over the smallest non-pointed basic Hopf algebra
• Some Hopf algebras of dimension 72 without the Chevalley property
• Examples of finite-dimensional Hopf algebras with the dual Chevalley property
• Lifting via cocycle deformation
• Finite-dimensional Pointed or Copointed Hopf algebras over affine racks
• The Nichols algebra of a semisimple Yetter-Drinfeld module
• Classification of arithmetic root systems
• Classification of semisimple Hopf algebras of dimension 16
• Lifting of Quantum Linear Spaces and Pointed Hopf Algebras of order p³

Related

• Finite dimensional Nichols algebras over H_c: σ₀ of Kashina
• On Nichols algebras over basic Hopf algebras
• On finite-dimensional Hopf algebras
• Some Hopf algebras of dimension 72 without the Chevalley property
• Biproducts and Kashina's examples