# Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups

## TL;DR

This paper classifies finite-dimensional pointed Hopf algebras over certain finite simple groups of Lie type, showing that most conjugacy classes do not support such algebras, except for the group algebra itself.

## Contribution

It proves that all non-semisimple, non-unipotent classes in specified Chevalley and Steinberg groups collapse, except for PSL_n(q), and identifies the only finite-dimensional pointed Hopf algebras over certain groups.

## Key findings

- All classes that are neither semisimple nor unipotent in specified groups collapse.
- The only finite-dimensional pointed Hopf algebra over these groups is the group algebra.
- Classification of finite-dimensional pointed Hopf algebras over certain finite simple groups.

## Abstract

We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from $PSL_n(q)$ collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is $PSp_{2n}(q)$, $P\Omega^+_{4n}(q)$, $P\Omega^-_{4n}(q)$, $^3D_4(q)$, $E_7(q)$, $E_8(q)$, $F_4(q)$, or $G_2(q)$ with q even is the group algebra.