Abstract
As the duality of generalized path algebra (see [4]), we firstly introduce the concept
of generalized path coalgebra through assigning a k-coalgebra to each vertex of a
given quiver. Some elementary properties of generalized path coalgebras are given.
Moreover, we discuss the isomorphism problem. It is shown that two generalized path
coalgebras are isomorphic with each other if and only if their quivers are isomorphic
with certain condition. For a coalgebra with CodimC0 · 1, the Wedderburn-Malcev
Theorem are given, that is, there exists a coalgebra projection of C onto C0. It is
a generalization of the Wedderburn-Malcev Theorem on coalgebras with separable
coradicals. As an important application of generalized path coalgebras, the Dual
Gabriel Theorem on a pointed coalgebra is generalized to a coalgebra with some
conditions, in particular, with separable coradical, so as to embedding such coalgebra
into a generalized path coalgebra from the quiver of the cotensor coalgebra of the
coalgebra. Lastly, we discuss the uniqueness of the quiver of the cotensor coalgebra
under the meaning of embedding. And, it is shown that the quiver is a wide subquiver
of another quiver defined by Montgomery[11].
1 Introduction
In this paper, we always suppose that k denotes a field and all linear spaces are over k.
The concept of generalized path algebra was introduced in [4], which is a generalization
of path algebra through assigning a k-algebra to each vertex of a given quiver. In [4], some
properties of generalized path algebras were given, including the so-called isomorphism
¤Project(No.102028) supported by the Natural Science Foundation of Zhejiang Province of China
yfangli@zju.edu.cn
zgxliu7969@sohu.com