Finite Dimensional Hopf Actions on Central Division Algebras

Let k be an algebraically closed field of characteristic zero. Let D be a division algebra of degree d over its center Z(D). Assume that k. Z(D). We show that a finite group G faithfully grades D if and only if G contains a normal abelian subgroup of index dividing d. We also prove that if a finite dimensional Hopf algebra coacts on D defining a Hopf-Galois extension, then its PI degree is at most d(2). Finally, we construct Hopf-Galois actions on division algebras of twisted group algebras attached to bijective cocycles.