| Abstract: |
| The theory of quiver representations over the virtual field $\mathbb{F}_1$ can be naturally extended to bound quivers involving monomial and commutativity relations. We further investigate Tits`s philosophy on $\mathbb{F}_1$ by examining the Ringel-Hall algebras associated with specific classes of bound quivers, including Nakayama bound quivers and gentle one-cycle quivers $\Lambda(n-1, 1, 1)$. Our findings extend and refine the results of Szczesny for quivers of type $A_n$ and cyclic quivers. |
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