Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III. Semisimple classes in PSL(n,q)

TL;DR

This paper investigates the structure of Nichols algebras over finite simple groups of Lie type, showing that most semisimple classes lead to infinite-dimensional algebras and identifying conditions under which pointed Hopf algebras are trivial.

Contribution

It introduces a new criterion for class collapse and proves that for many (n,q), finite-dimensional pointed Hopf algebras over PSL(n,q) are just group algebras.

Findings

Most Nichols algebras over semisimple orbits are infinite-dimensional.

A new criterion for conjugacy class collapse is established.

Finite-dimensional pointed Hopf algebras over PSL(n,q) are often group algebras.

Abstract

We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to semisimple orbits, have infinite dimension. We introduce a new criterium to determine when a conjugacy class collapses and prove that for infinitely many pairs (n,q), any finite-dimensional pointed Hopf algebra H with G(H) = PSL(n,q) or SL(n,q) is isomorphic to a group algebra.