The Hochschild Cohomology of the Weak Entwining Structure of a Path Algebra and a Path Coalgebra

In order to investigate the properties of weak entwining structures, in this paper we firstly use the methods of homological algebra to discuss the weak entwining structure of a path algebra and a path co-algebra, and then we research the Hochschild cohomology of the weak entwining structure and its properties, and obtain equivalent conditions that the underlying graph of the path coalgebra in such a weak entwining structure is a tree.