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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 , any finite-dimensional pointed Hopf algebra with or is isomorphic to a group algebra.
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Nicolás Andruskiewitsch, Giovanna Carnovale, Gastón Andrés García, Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type III. Semisimple classes in PSL. Rev. Mat. Iberoam. 33 (2017), no. 3, pp. 995–1024