Green Rings Of A Family Of Pointed Hopf Algebras Of Rank Two

The tensor product of representations of a Hopf algebra is an important ingredient in the representation theory of Hopf algebra and quantum groups.In particular,the decomposition of the tensor product of indecomposable modules into direct sum of indecomposables has received enormous attention.However,in general,very little is known about how a tensor product of two indecomposable representations decomposes into a direct sum of indecomposable representations.One method of addressing this problem is to consider the tensor product as the multiplication of the Green ring(or the representation ring)and study the ring structure of the Green ring.On the other hand,cocycle deformation is one of the main methods in studying the structures and classifications of Hopf algebras.However,very little is known about the relations between the monoidal categories of the modules over two cocycle twist-equivalent Hopf algebras.In this thesis,we study the tensor product decomposition rules for finite dimen-sional indecomposable modules over the Drinfeld doubles D(Hn(q))≌Hn(1,q)of Taft Hopf algebras Hn(q),and the Green rings of the Drinfeld doubles Hn(1,q).We also study the projective class rings of the tensor product Hn(q)=Hn(q-1)(?)Hn(q)of Taft algebras Hn(q-1)and Hn(q),and its two cocycle deformations Hn(0,q)and Hn(1,q),where N>2 is a positive integer and Q is an nth primitive root of ulity in the ground field.Firstly,we investigate the tensor product of two finite dimensional indecom-posable modules over Hn(1,q),decompose such tensor products into a direct sum of indecomposable modules.The tensor product decomposition rules for finite di-mensional indecomposable modules over Hn(1,q)are all explicitly given.Then we investigate the projective class ring and the Green ring(or the representation ring)of Hn(1,q).The projective class ring and the Green ring of Hn(1,q)are described in terms of generators and relations.Finally,we investigate the representations and the projective class rings of Hn(q)and Hn(0,q).The representation types of Hn(q)and Hn(0,q)are determined.We classify the simple modules and indecomposable projective modules over Hn(q)and Hn(0,q),and decompose the tensor products of these modules into a direct sum of indecomposable modules.This leads the descrip-tion of the projective class ring,the Jacobson radical of the projective class algebra and the corresponding quotient algebras for Hn(q)and Hn(0,q),respectively.It turns out that even the Hopf algebras Hn(q),Hn(0,q)and Hn(1,q)are cocycle twist-equivalent to each other,they are of different representation types:wild,wild and tame,respectively.Moreover,they own the different number of blocks with 1,n and n(n+1)/2,respectively.They also own the different Loewy lengths:2n-1,2n-1 and 3,respectively.Hn(q)and Hn(0,q)are basic algebras,but Hn(1,q)is not.Hn(0,q)and Hn(1,q)are symmetric algebras,but Hn(q)is not.Furthermore,the projective class rings of Hn(q),Hn(0,q)and Hn(1,q)are piecewise non-isomorphic.