Representations of Quivers over F1

TL;DR

This paper develops a theory of quiver representations over the field with one element, defining a Hall algebra and establishing connections to Kac-Moody algebras, extending classical representation theory concepts to this new setting.

Contribution

It introduces the category of quiver representations over F1, constructs its Hall algebra, and links it to Kac-Moody algebra structures, providing new insights into algebraic representations over F1.

Findings

Defined the category of quiver representations over F1.

Constructed the Hall algebra of these representations.

Established a homomorphism from Kac-Moody algebra to the Hall algebra.

Abstract

We define and study the category $\RepQ$ of representations of a quiver in $\VFun$ - the category of vector spaces "over $\Fun$". $\RepQ$ is an $\Fun$-linear category possessing kernels, co-kernels, and direct sums. Moreover, $\RepQ$ satisfies analogues of the Jordan-H\"older and Krull-Schmidt theorems. We are thus able to define the Hall algebra $\HQ$ of $\RepQ$, which behaves in some ways like the specialization at $q=1$ of the Hall algebra of $\on{Rep}(\Q, \mathbf{F}_q)$. We prove the existence of a Hopf algebra homomorphism of $ \rho': \U(\n_+) \rightarrow \HQ$, from the enveloping algebra of the nilpotent part $\n_+$ of the Kac-Moody algebra with Dynkin diagram $\bar{\Q}$ - the underlying unoriented graph of $\Q$. We study $\rho'$ when $\Q$ is the Jordan quiver, a quiver of type $A$, the cyclic quiver, and a tree respectively.