Hopf algebras over Chevalley groups

TL;DR

This paper proves that finite-dimensional pointed Hopf algebras over most finite simple Chevalley groups are isomorphic to their group algebras, extending Nichols algebra analysis and introducing new criteria for conjugacy class types.

Contribution

It completes the classification of pointed Hopf algebras over Chevalley groups by analyzing Nichols algebras and developing new methods for conjugacy class classification.

Findings

Most finite simple Chevalley groups have trivial pointed Hopf algebras beyond group algebras.

Introduced a general procedure to determine when a conjugacy class is of type C.

Extended Nichols algebra results to racks beyond Chevalley groups.

Abstract

We show that every finite-dimensional pointed Hopf algebra over a finite simple Chevalley group, different from $PSL_2(q)$ with q= 3 mod 4 (and from $PSL_3(2)\simeq PSL_2(7)$), is isomorphic to the corresponding group algebra. To do this, we complete the analysis of the Nichols algebras of Yetter-Drinfeld modules over such groups whose support is a semisimple orbit, begun in arXiv:1506.06794, arXiv:2301.03361. In addition to the techniques used in loc. cit., we introduce a general procedure to determine when a semisimple conjugacy class in a Chevalley or Steinberg group is of type C and a new criterion based on the results of arXiv:2411.02304 that applies to arbitrary racks. Throughout the process, we obtain results on Nichols algebras over racks beyond the framework of Chevalley groups.