Representations of Quivers over F1

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