Finite dimensional Hopf actions on central division algebras

Abstract

Let $\mathbb{k}$ be an algebraically closed field of characteristic zero. Let $D$ be a division algebra of degree $d$ over its center $Z(D)$. Assume that $\mathbb{k}\subset Z(D)$. 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.

14 pages. To appear in Int. Math. Res. Not. Remarks 2.9 and 5.4 are new. Another proof of Lemma 2.8 was given. Some minors changes on writing

References in corpus (3)

• Semigroups of I-type
• Finite dimensional graded simple algebras
• "Set-theoretical" solutions of the quantum Yang-Baxter equation and a class of Garside groups