Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSL(q) and PSp2(q)

Abstract

We show that the Nichols algebra of a simple Yetter-Drinfeld module over a projective special linear group over a finite fieldwhose support is a semisimple orbit has infinite dimension, provided that the elements of the orbit are reducible; weobtain a similar result for all semisimple orbits in a finite symplectic group except in low rank. We prove that orbitsof irreducible elements in the projective special linear groups of odd prime degree could not be treated with our methods.We conclude that any finite-dimensional pointed Hopf algebra H with group of group-like elements isomorphic to PSLn(q)(n ≥ 4), PSL3(q) (q > 2), or PSp2n(q) (n ≥ 3), is isomorphic to a group algebra.