Generalized Path Algebras and Their Hochschild Cohomology

To study a generalized path algebra on the quiver Δ,all its indecomposable projective modules and injective modules,and simple modules were constructed by using a complete set of its orthogonal primitive elements. Based on the properties of hereditary algebras a sufficient and necessary condition for a generalized path algebra R = k( Δ,A) to be hereditary was proposed. Furthermore,the Hochschild cohomology of R = k( Δ,A) based on homology theory and Hochschild cohomology of a finitely dimensional algebra was obtained.