Characterization Of Left Artinian Algebras Through Pseudo Path Algebras

TL;DR

This paper extends Gabriel's Theorem to left Artinian algebras over a field using pseudo path algebras, providing new structural characterizations and uniqueness results for these algebras.

Contribution

It introduces pseudo path algebras as a new tool to generalize Gabriel's Theorem for left Artinian and finite dimensional algebras with specific radical properties.

Findings

Characterization of left Artinian algebras via pseudo path algebras

Isomorphism conditions involving relations and radicals

Uniqueness of quivers and algebra representations under admissibility

Abstract

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).