Representations of Quivers Over 𝔽1 and Hall Algebras

We define and study the category $\mathrm Rep(Q,\ \mathbb F_1)$ of representations of a quiver in $\mathrm Vect(\mathbb F_1)$—the category of vector spaces “over $\mathbb F_1$.” $\mathrm Rep(Q,\ \mathbb F_1)$ is an $\mathbb F_1$-linear category possessing kernels, co-kernels, and direct sums. Moreover, $\mathrm Rep(Q,\ \mathbb F_1)$ satisfies analogues of the Jordan–Hölder and Krull–Schmidt theorems. We are thus able to define the Hall algebra HQ of $\mathrm Rep(Q,\ \mathbb F_1)$, 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 $\rho ':{\mathrm U}(\mathfrak n_+)\rightarrow \mathrm {H_Q}$, from the enveloping algebra of the nilpotent part $\mathfrak n_+$ of the Kac–Moody algebra with Dynkin diagram $\bar {\mathrm Q}$—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.