Representations of Quivers over F1
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, cokernels, and direct sums. Moreover, $\RepQ$ satisfies analogues of the JordanHölder and KrullSchmidt 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 KacMoody 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2010
 DOI:
 10.48550/arXiv.1006.0912
 arXiv:
 arXiv:1006.0912
 Bibcode:
 2010arXiv1006.0912S
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Combinatorics;
 Mathematics  Representation Theory