The Hopf Automorphism Group of Two Classes of Drinfeld Doubles

Let D(R-m,R-n(q)) be the Drinfeld double of Radford Hopf algebra R-m,R-n(q) and D(H-s,H-t) be the Drinfeld double of generalized Taft algebra H-s,H-t. Both D(R-m,R-n(q)) and D(H-s,H-t) have very symmetric structures. We calculate all Hopf automorphisms of D(R-m,R-n(q)) and D(H-s,H-t), respectively. Furthermore, we prove that the Hopf automorphism group Aut(Hopf)(D(R-m,R-n(q))) is isomorphic to the direct sum Z(n) circle plus Z(m) of cyclic groups Z(m) and Z(n), the Hopf automorphism group Aut(Hopf)(D(H-s,H-t)) is isomorphic to the semi-direct products k* (sic) Z(d) of multiplicative group k* and cyclic group Z(d), where s=td, k*=k\{0}, and k is an algebraically closed field with char (k) inverted iota t.