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 [Formula: see text], the quotient of the non-pointed prime Hopf algebras of GK-dimension one, which is generated by [Formula: see text] with the relations: [Formula: see text] where [Formula: see text] is a [Formula: see text]th primitive root of unity. First, we describe the group formed by permutations of the isomorphism classes of indecomposable modules of [Formula: see text], which can be extended to automorphisms of the representation ring of [Formula: see text]. An element within this group is regarded as a permutation of the set of points of AR-quiver of [Formula: see text] 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 [Formula: see text]. By straightforward computation, it is shown that the automorphism group of the representation ring of [Formula: see text] is isomorphic to Klein four group [Formula: see text].