Representations of Bound Quivers over ${\mathbb{F}}_1$ and Hall Algebras

Abstract

Representations of quivers over the virtual field ${\mathbb{F}}_1$ can be naturally extended to bound quivers with monomial and commutativity relations. In this paper, we introduce and study the Hall algebra ${\mathcal{H}}_{{\mathbb{F}}_1}(Q,I)$ for a bound quiver $(Q,I)$ equipped with such relations. As in the quiver setting, ${\mathcal{H}}_{{\mathbb{F}}_1}(Q,I)$ is a graded, connected, and cocommutative bialgebra, and the free subgroup generated by all indecomposable representations forms a Lie subalgebra. We study ${\mathcal{H}}_{{\mathbb{F}}_1}(Q,I)$ in detail for two special classes of bound quivers: Nakayama bound quivers and gentle one-cycle bound quivers. In both cases, we prove that for any triple $L,M,N$ of ${\mathbb{F}}_1$-representations over $(Q,I)$, the Hall number $g_{N,L}^M$ coincides with the evaluation of the corresponding Hall polynomial ${\varphi }_{N,L}^M(t)$ at $t=1$. Consequently, there exists a canonical isomorphism between ${\mathcal{H}}_{{\mathbb{F}}_1}(Q,I)$ and the degenerate Hall algebra ${\mathcal{H}}_1(Q,I)$, justifying Tits's ${\mathbb{F}}_1$-philosophy in these settings.

arXiv comment28 pages, 2 figures