Polynomial identities for quivers via incidence algebras

TL;DR

This work shows that the path algebra $FQ$ of a quiver $Q$ has the same polynomial identities as an incidence algebra whenever $Q$ is a PI quiver, establishing a PI-equivalence between $FQ$ (or its path-generated subalgebras $FQ_\pi$) and incidence algebras $A_\pi$. The authors construct a map $\varphi_Q: FQ\to M_n(F)$ with $A_Q=\mathrm{im}(\varphi_Q)$ and $A_\pi=\varphi_Q(FQ_\pi)$, and prove $\mathrm{Id}(FQ_\pi)=\mathrm{Id}(A_\pi)$ for all $\pi$ when $Q$ is PI; in particular, $\mathrm{Id}(FQ)=\mathrm{Id}(A_Q)$. This yields explicit instances where incidence algebras share identities with matrix algebras, notably $A_Q\cong M_n(F)$ for the oriented cycle $Q=C_n$, giving $\mathrm{Id}(FC_n)=\mathrm{Id}(M_n(F))$. The results connect polynomial identities of quiver path algebras with the well-studied theory of incidence algebras and provide a broad family of PI algebras tied to matrix identities.

Abstract

We show that the path algebra of a quiver satisfies the same polynomial identities of an algebra of matrices, if any. In particular, the algebra of nxn matrices is PI-equivalent to the path algebra of the oriented cycle with n vertices.

Theorems & Definitions (6)

• Theorem 1
• Corollary 2
• Corollary 3
• Example 4
• Example 5
• Example 6