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Hopf algebras over Chevalley groups

Nicolás Andruskiewitsch, Giovanna Carnovale

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.

Hopf algebras over Chevalley groups

Abstract

We show that every finite-dimensional pointed Hopf algebra over a finite simple Chevalley group, different from with q= 3 mod 4 (and from ), 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.
Paper Structure (42 sections, 70 theorems, 185 equations, 4 tables)

This paper contains 42 sections, 70 theorems, 185 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Pointed Hopf algebras
  3. Nichols algebras over racks
  4. The strategy
  5. Notation
  6. Nichols algebras over racks
  7. Kthulhu conjugacy classes
  8. Detecting type C via quasi-simple subgroups
  9. Solvable racks
  10. A new criterion on Nichols algebras
  11. Propagation
  12. Preliminaries on finite groups of Lie type
  13. Groups of Lie type
  14. Semisimple conjugacy classes
  15. Cuspidal classes in Weyl groups of type and
  16. ...and 27 more sections

Key Result

Theorem 1.3

The finite simple Chevalley groups, other than $\mathbf{PSL}_2(q)$ with $q \equiv 3 \mod 4$ (and than $\mathbf{PSL}_3(2)\simeq\mathbf{PSL}_2(7)$), collapse.

Theorems & Definitions (154)

  • proof
  • Conjecture 1.2
  • Theorem 1.3
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 2.1
  • Lemma 2.2
  • Theorem 2.3
  • ...and 144 more