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 $\{e_{ii}\}_I$ with each $\bar e_{ii}$ in the center of $\bar Λ= Λ/r$ 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 characteristic 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< \infty $; $Λ$ 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.

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