Hopf algebras over Chevalley groups

Abstract

We show that every finite-dimensional pointed Hopf algebra over a finite simple Chevalley group, different from $PS{L}_2(q)$ with q= 3 mod 4 (and from $PS{L}_3(2)\simeq PS{L}_2(7)$), 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.

80 pages