Automorphism groups associated to a family of Hopf algebras

In this paper, we discuss the group consisting of some good automorphisms of representation ring of the non-pointed Hopf algebra D(n)D(n), the quotient of the non-pointed prime Hopf algebras of GK-dimension one, which is generated by x,y,zx,y,z with the relations: x(2)n=1,y(2)=0,z(2)=xn,xy=-yx, xz=zx(-1), yz=omega zy, where omega omega is a 44th primitive root of unity. First, we describe the group formed by permutations of the isomorphism classes of indecomposable modules of D(n)D(n), which can be extended to automorphisms of the representation ring of D(n)D(n). An element within this group is regarded as a permutation of the set of points of AR-quiver of D(n)D(n) such that the tensor product of indecomposable modules corresponding to these points are isomorphic. Then, we try to understand automorphism group of representation ring of D(n)D(n). By straightforward computation, it is shown that the automorphism group of the representation ring of D(3)D(3) is isomorphic to Klein four group K4K4.