The Structure Of Hopf Algebras On Path Coalgebras

When H.Hopf studied the homology of compact Lie group, he gave the concept of Hopf algebra. We knew that it is extensively connected with Lie algebra, differential geometry, algebraic topology and statistical physics. One got many important results about constucting and classifying Hopf algebras over the past several decades. We try to find a class of non-commutative non-cocommutative Hopf algebra, then to study the structure properties and representation properties of them. In order to get some concrete examples of quantum groups, it is important to investigate a constructing method of Hopf algebras. Several methods are known to constuct a quantum groups now, such as Ringel-Hall algebra approach, FRT method , the Lusztig's geometry method, the Drinfeld-Jimbo method, and so on. There are many methods to constuct a Hopf algebra, such as the method of dual of Hopf algebra, the method of group algebra, the extention method of Hopf algebra, the method of Drifeld Double, the method of Hopf quiver, and etc. In this paper, we apply Cibils and Rosso's method to construct a Hopf algebra on a given Hopf quiver.In this thesis, we give the background and main results, and the properties of Hopf quiver. A criteria for a quiver, which is coresponding to a cyclic group, to be a Hopf quiver, is given. By the method of Cibils and Rosso to construct a Hopf algebra kQ(a)~c on Q(a), we construct a deformed preprojective algebra Π~q(Q(a)) over Q(a), which has a Hopf algebra structure. We also describe the...