Mixed quiver algebras

TL;DR

This paper introduces a new class of $K$-algebras associated with quivers, called mixed path, Leavitt path, and regular path algebras, extending classical constructions to a layered field and hereditary subset framework.

Contribution

It constructs and analyzes mixed path, Leavitt path, and regular path algebras for quivers, generalizing existing algebraic structures with layered field and hereditary subset data.

Findings

Shared many properties with classical algebras

Extended algebraic structures to layered fields and hereditary subsets

Provided foundational definitions and properties for new algebra classes

Abstract

In this paper we introduce a new class of $K$-algebras associated with quivers. Given any finite chain $\mathbf{K}_r: K=K_0\subseteq K_1\subseteq ... \subseteq K_r$ of fields and a chain $\mathbf{E}_r : H_0\subset H_1\subset ... \subset H_r=E^0$ of hereditary saturated subsets of the set of vertices $E^0$ of a quiver $E$, we build the mixed path algebra $P_{\mathbf{K}_r}(E,\mathbf{H}_r)$, the mixed Leavitt path algebra $L_{\mathbf{K}_r}(E,\mathbf{H}_r)$ and the mixed regular path algebra $Q_{\mathbf{K}_r}(E,\mathbf{H}_r)$ and we show that they share many properties with the unmixed species $P_K(E)$, $L_K(E)$ and $Q_K(E)$.