The Hopf Automorphism Group of Two Classes of Drinfeld Doubles

Abstract

Let D(Rm,n(q)) be the Drinfeld double of Radford Hopf algebra Rm,n(q) and D(Hs,t) be the Drinfeld double of generalized Taft algebra Hs,t. Both D(Rm,n(q)) and D(Hs,t) have very symmetric structures. We calculate all Hopf automorphisms of D(Rm,n(q)) and D(Hs,t), respectively. Furthermore, we prove that the Hopf automorphism group AutHopf(D(Rm,n(q))) is isomorphic to the direct sum Zn⨁Zm of cyclic groups Zm and Zn, the Hopf automorphism group AutHopf(D(Hs,t)) is isomorphic to the semi-direct products k*⋊Zd of multiplicative group k* and cyclic group Zd, where s=td,k*=k\{0}, and k is an algebraically closed field with char (k)∤t.

Keywords

• Hopf algebra
• Drinfeld double
• automorphism group