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We show that Nichols algebras of most simple Yetter-Drinfeld modules over theprojective special linear group over a finite field, corresponding tosemisimple orbits, have infinite dimension. We introduce a new criterium todetermine when a conjugacy class collapses and prove that for infinitely manypairs (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.

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