Automorphism Group Of Hopf Algebra H(l,q)

Hopf algebra is an important topic in algebra.In the past,many mathematicians have been working on the structures and classification of Hopf algebras,and have great achievement.Hopf algebras are closely related to quantum groups,Lie theory,representation theory and so on.It is well known that the automorphism group of an algebraic structure reflects the symmetry of the structure,and that the group is helpful for us to understand the structure.The study of automorphisms and automorphism groups of Hopf algebras is a fundamental work.The automorphism group is also one of the important tools in the study of Hopf algebras.In this paper,we study the Hopf algebra automorphisms and automorphism groups of a family of Hopf algebras.In 1999,Chen constructed a family of Hopf algebras H(p,q),where p,q are scalars in the ground field k with q?0.When q is an nth primitive root of unity for some integer n?2,H(p,q)has a quotient Hopf algebra Hn(p,q)of dimension n4.Furthermore,if p?0 then Hn(p,q)is isomorphic to the Drinfeld quantum double of the Taft algebra of dimension n4.Based on previous studies,we study the automorphisms and the automorphism groups of Hopf algebra H(1,q).This paper is organized as follows.In Section 1,we recall some basic concepts,such as Hopf algebra,Hopf algebra isomorphism,semidirect product and so on.We also recall the structure of H(1,q).In Section 2,we give several families of Hopf algebra automorphisms on H(1,q)for q=1,q=-1(when the characteristic of the groud field k is not 2)and q?± 1,respectively.In Section 3,we first recall some properties of H(1,q).Then we investigate some basic properties of Hopf algebra automorphisms on H(1,q).We find that the Hopf algebra automorphisms on H(1,q)have different expressions for different values of the parameter q.Finally,for q=1,q=-1 and q?±1,we study the Hopf algebra automorphisms on H(1,q),respectively.It is shown that the automorphisms given in Section 2 are exactly all Hopf algebra automorphisms on H(1,q).Moreover,we describe the structures of automorphism groups Aut(H(1,q))for different q.When q=1,it is shown that the automorphism group Aut(H(1,1))of H(1,1)fits into the short exact sequence 1?k*(?)kk2 ?Aut(H(1,1))?Z2?1 and that Aut(H(1,1))is isomorphic to the quotient of the semidirect product group(k*(?)K2)(?)Z4 modulo a cyclic normal subgroup of order 2,where k*is the multiplicative group of nonzero scalars in k,k2 is the additive group k(?)k and Z4 is the cyclic group of order 4.When q=-1 it is shown that Aut(H(1,-1))is isomorphic to the semidirect product group k*×Z2,where Z2 is the cyclic group of order 2.When q?± 1,it is shown that Aut(H(1,q))is isomorphic to the multiplicative group k*of nonzero scalars.Finally,it is pointed out that the automorphism group Aut(H(1,q))can be gotten from the disscussion before when char(k)=2.