$n$-representations of Quivers

TL;DR

This paper introduces the concept of n-representations of quivers, establishing their categorical properties and demonstrating that these categories are k-linear abelian, thus extending classical quiver representation theory.

Contribution

It defines n-representations of quivers, constructs their categories, and proves these categories are k-linear abelian, providing a new framework for higher-dimensional quiver representations.

Findings

Categories of n-representations are k-linear abelian.

Every morphism in n-representations has a canonical decomposition.

Constructs kernels and cokernels in n-representations using classical quiver categories.

Abstract

Let $n \geq 2$. We introduce the notion of $n$-representations of quivers, and we explicitly provide concrete examples of $2$-representations of quivers. We establish the categories of $n$-representations and investigate kernels and cokernels in the categories of $n$-representations of quivers. Further, we construct them in terms of kernels and cokernels of morphisms in the usual categories of quiver representations. We show that every morphism in the categories of $n$-representations has a canonical decomposition. Most importantly, we prove that the categories of $n$-representations of quivers are $k$-linear abelian categories.