On Cohomology Of Cyclic Path Algebras And Their Quotient Algebras

The present thesis mainly concerns the cohomology theory of path alge-bras and their quotient algebras. Besides, we investigate the properties of path categories and complete path algebras.Firstly, we consider the graded path category associated to a quiver. We investigate all n-differentials on such a category, and also study the associated graded Lie algebra. Moreover, a sufficient and necessary condition is given that ensures the graded path category admits a DG category structure.Secondly, we characterize the first graded Hochschild cohomology of a hered-itary algebra whose Gabriel quiver is admitted to have oriented cycles. The in-teresting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra.Thirdly, we study the first cohomology of admissible algebras which can be seen as a generalization of basic algebras. Differential operators on an admissible algebra are studied. Based on the discussion, the dimension formula of the fist cohomology of admissible algebras is characterized. In particular, for planar quivers, the linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field k of characteristic0.At last, we study the cohomology of complete path algebras. Complete path algebras can be seen as an inverse limit of a sequence of truncated path algebras. Due to this view, we can adopt the method of studying the cohomology of profinite groups. The conclusion we get shows that the cohomology of a complete path algebra with a discrete bimodule as coefficient is a direct limit of a sequence of the cohomology of truncated path algebras.