Structures and Representations of Generalized Path Algebras

It is shown that an algebra Λ can be lifted with nilpotent Jacobson radical r = r(Λ) and has a generalized matrix unit {eii}I with each ēii in the center of Λ̄ = Λ/r iff Λ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, Λ is a finite algebra with non-zero unity element over perfect field k (e.g. a field with character zero or a finite field ) iff Λ is isomorphic to a generalized path algebra k(D,Ω, ρ) of finite directed graph with weak relations and dim Ω < ∞; Λ is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise orthogonal idempotents iff Λ is isomorphic to a path algebra with relations.