The Automorphism Groups Of Two Grothendieck Rings

Since the 1980s,with the rise of quantum groups,many mathematicians have devoted themselves to the research of Hopf algebra.The concept of a quasitriangular Hopf algebra was introduced by Drinfeld when he studied the quantum Yang-Baxter equations.The concept is important since the representations of a quasitriangular Hopf algebra can provide solutions to the quantum Yang-Baxter equations.Drinfeld also gave a general method to construct a quasitriangular Hopf algebra D(H)from a finite-dimensional Hopf algebra H,which is now called the Drinfeld double of H.Taft algebras are a family of finite dimension Hopf algebra invented by Taft in 1972.The relation of the two generators of Taft algebras plays a role of inspiration when many quantum groups and Hopf algebras are being constructed.Invariants are important tools in the study of the structures of algebras.Hopf algebras have many invariants,for instance,automorphism group,Green ring,Grothendieck ring and so on.Automorphism group is an important invariant that reflects the symmetry of algebra.However,there is no known method to determine the automorphism group of a ring or an algebra.In 1999,Chen constructed a family of finite-dimensional Hopf algebras Hn(p,q),where n>1 is an integer and q is a primitive n-th root of unity.When p≠0,Hn(p,q)is exactly isomorphic to the Drinfeld doubles of Taft algebras.Many results have been gotten around the structure and representation theory of the Drinfeld doubles of Taft algebras.In 2008,Zhang,Wu,Liu and Chen described the ring structure of the Grothendieck group G0(Hn(1,q))of the Drinfeld double of the Taft algebra.It is shown that G0(Hn(1,q))is isomorphic to a quotient ring of the polynomial ring in two variablesBased on the work of the previous,we continue the study of the Grothendieck rings of the Drinfeld doubles of Taft algebra in this thesis.We mainly focus on the automorphism groups of Grothendieck rings G0(H2(1,q))and G0(H3(1,q)).This thesis is organized as follows.In Chapter one,we introduce the concepts of the Grouthendieck ring of a Hopf algebra and the Drinfeld doubles Hn(1,q)of Taft algebras,and the structure of the Grouthendieck ring G0(Hn(1,q)).In Chapter two,we study the automorphism group of the Grouthendieck ring G0(H2(1,q))of the Drinfeld double of the 4-dimensional Taft algebra Firstly,we construct three additive group automorphisms of G0(H2(1,q)).It is shown that they are all ring automorphisms of G0(H2(1,q)),and that they,together with the identity map on G0(H2(1,q)),form a subgroup of the automorphism group of the ring G0(H2(1,q))Meanwhile,it is proved that this subgroup is isomorphic to Kleinian group.Then in two different ways,we show that any ring automorphism of G0(H2(1,q))is one of the elements in the subgroup described above.Therefore,the ring automorphism group of G0(H2(1,q))is isomorphic to Kleinian group.In Chapter 3,we study the ring automorphism of the Grothendieck ring G0(H3(1,q))of the Drinfeld double of the 9-dimensional Taft algebra First,we construct an additive group automorphism of G0(H3(1,q))and prove that it is a ring automorphism of order 2.Therefore,the ring automorphism generates a cyclic subgroup of order 2 of the ring automorphism group of G0(H3(1,q)).Then it is proved that a ring automorphism of G0(H3(1,q))is either the identity map on Go(H3(1,q))or the automorphism constructed above.This shows that the ring automorphism group of G0(H3(1,q))is isomorphic to the cyclic group of order 2.