Classification of PM Quiver Hopf Algebras

We give the classification of quiver Hopf algebras with pointed module structures. This leads to the classification of multiple crown algebras and multiple Taft algebras as well as pointed Yetter-Drinfeld modules and their corresponding Nichols algebras. In particular, we give the classification of graded Hopf algebras on cotensor coalgebra T c kG(M) of kG-bicomdule M over finite commutative group algebra kG and tensor algebra TkG(M) of dimensional finite kG bimodule M over finite commutative group algebra kG with |G |= n, when k contains all roots of all irreducible polynomials over k with degree ≤ n and the order of G is not divisible by char(k). In this case, we also give the classification of Yetter-Drinfeld kG-modules and Nichols algebras of Yetter-Drinfeld kG-modules. 2000 Mathematics Subject Classification: 16W30, 16G10 keywords: Quiver, Hopf algebra, Hopf bimodule