Characterization Of Left Artinian Algebras Through Pseudo Path Algebras

In this paper, using pseudo path algebras, we generalize Gabriel's Theorem on elementary algebras to left Artinian algebras over a field $k$ when it is splitting over its radical, in particular, when the dimension of the quotient algebra decided by the $n$'th Hochschild cohomology is less than 2 (for example, $k$ is finite or char$k=0$). Using generalized path algebras, the generalized Gabriel's Theorem is given for finite dimensional algebras with 2-nilpotent radicals which is splitting over its radical. As a tool, the so-called pseudo path algebras are introduced as a new generalization of path algebras, which can cover generalized path algebras (see Fact 2.5). The main result is that (i) for a left Artinian $k$-algebra $A$ and $r=r(A)$ the radical of $A$, when the quotient algebra $A/r$ can be lifted, it holds that $A\cong PSE_k(\Delta,\mathcal{A},\rho)$ with $J^{s}\subset<\rho>\subset J$ for some $s$ (Theorem 3.2); (ii) for a finite dimensional $k$-algebra $A$ with $r=r(A)$ 2-nilpotent radical, when the quotient algebra $A/r$ can be lifted, it holds that $A\cong k(\Delta,\mathcal{A},\rho)$ with $\widetilde J^{2}\subset<\rho>\subset\widetilde J^{2}+\widetilde J\cap$ \textrm{Ker}$\widetilde{\varphi}$ (Theorem 4.3), where $\Delta$ is the quiver of $A$ and $\rho$ is a set of relations. Meantime, the uniqueness of the quivers and generalized path algebra/pseudo path algebras satisfying the isomorphism relations is obtained in the case when the ideals generated by the relations are admissible (see Theorem 3.5 and 4.4).