Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III. Semisimple classes in PSL${}_n(q)$

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 $G(H)\simeq {\mathbf{PSL}}_n(q)$ or ${\mathbf{SL}}_n(q)$ is isomorphic to a group algebra.