In this paper, we intend to study idempotents of the Green algebra (complexified Green ring) of any finite dimensional pointed rank one Hopf algebra of nilpotent type over the complex number field. We first determine all one dimensional representations of the quotient algebra of the Green algebra modulo its Jacobson radical. | Turk J Math (2016) 40: 789 – 805 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Idempotents of the Green algebras of finite dimensional pointed rank one Hopf algebras of nilpotent type Zhihua WANG∗ Department of Mathematics, Nanjing Normal University Taizhou College, Taizhou, . China Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, we intend to study idempotents of the Green algebra (complexified Green ring) of any finite dimensional pointed rank one Hopf algebra of nilpotent type over the complex number field. We first determine all one dimensional representations of the quotient algebra of the Green algebra modulo its Jacobson radical. This gives rise to all primitive idempotents of the quotient algebra. Then we present explicitly primitive idempotents of the Green algebra by lifting the ones of the quotient algebra. Finally, as an example, we describe all primitive idempotents of the Green algebra of the Taft algebra T3 . Key words: Hopf algebra, Green algebra, idempotent, Dickson polynomial 1. Introduction As we all know, the finite dimensional module category of any finite dimensional Hopf algebra is a tensor category, where the tensor product of any two indecomposable modules can always be decomposed as a direct sum of indecomposable modules. However, there is a less obvious way to obtain this decomposition. One method of addressing this problem is to consider the tensor product as the multiplication of the Green ring (or the representation ring), and to study the ring-theoretical properties of the Green ring, due to J. A. Green for the study of the modular representations of a finite group [4]. A lot of work have been done in this direction, see . [1, 3, 12, 18]. According to the Krull–Schmidt theorem, one knows that the Green ring of a Hopf algebra is a free Z -module with the isomorphism classes of
