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In the theory of representation, how to break up a tensor product of two indecomposable representation is of great importance. The Green ring is much more complicatied than the Grethenduick ring. One method of addressing this information is to consider the tensor product as the produt in the Green ring. Pointed Hopf algebra of rank one of nilpotent type is a algebra which aroused great interests of so many algebraists. In this paper, we take into account the algebra Hn,m(q,p) which is a special type of pointed Hopf algebra of rank one. As an associate algebra, Hn,m(q,p) is generated by three generators a, b, h as follows:an=1,bm=1,hn=0,ha=qah,hb=pbh,While q is an n-primitive root of unit, p is an. m-primitive root of unit.Concretely, in section1and section2, we present the research background and some relative conclusions of quantum algebra and Hopf algebra.In section3, we introduce some relative conclusions on pointed Hopf algebra of rank one. Based on these, we take account to the subject of this paper:Hn,m(q,p). It’s an associate algebra generated by three elements a,b,h as follows:an=1, bm=1, hn=0,ha=qah,hb=pbh, where q is a n-th primitive root of unit, pis a m-th primitive root of unit.In section4, we present all the non-isomorphism indecomposable modules of Hn,m (q, p) and break the tensor product of either two into a direct some of indecomposable modules.In section5, firstly, we prove the Green of Hn,m(q,p); secondly, we discuss the roots of Fibonacci polynomial in complex field. Lastly, we compute the rank of the Jacobson radical and the nilpotent elements in the Green ring.
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