Representations of Quivers over F1 and Hall Algebras

We define and study the category Rep(Q, F1) of representations of a quiver in Vect(F1) the category of vector spaces ”over F1”. Rep(Q, F1) is an F1–linear category possessing kernels, co-kernels, and direct sums. Moreover, Rep(Q, F1) satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra HQ of Rep(Q, F1), which behaves in some ways like the specialization at q = 1 of the Hall algebra of Rep(Q, Fq). We prove the existence of a Hopf algebra homomorphism of ρ′ : U(n+)→ HQ, from the enveloping algebra of the nilpotent part n+ of the Kac-Moody algebra with Dynkin diagram Q the underlying unoriented graph of Q. We study ρ′ when Q is the Jordan quiver, a quiver of type A, the cyclic quiver, and a tree respectively.