Finite-dimensional pointed Hopf algebras over finite 3 simple groups of Lie type V. Mixed classes in Chevalley 4 and Steinberg groups

Abstract

We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from PSLn(q) collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is PSp2n(q) , PΩ4n+(q), PΩ4n-(q), 3D4(q) , E7(q) , E8(q) , F4(q) , or G2(q) with q even is the group algebra.