On the Hall algebra of semigroup representations over $\mathbb F _1$

Abstract

Let $\mathrm{A }$ be a finitely generated semigroup with 0. An $\mathrm{A }$-module over $\mathbb F _1$ (also called an $\mathrm{A }$-set), is a pointed set $(M,*)$ together with an action of $\mathrm{A }$. We define and study the Hall algebra $\mathbb H _{\mathrm{A }}$ of the category $\mathcal C _{\mathrm{A }}$ of finite $\mathrm{A }$-modules. $\mathbb H _{\mathrm{A }}$ is shown to be the universal enveloping algebra of a Lie algebra $\mathfrak n _{\mathrm{A }}$, called the Hall Lie algebra of $\mathcal C _{\mathrm{A }}$. In the case of $\langle t \rangle$—the free monoid on one generator $\langle t \rangle$, the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\langle t \rangle$-modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\mathrm{A }$ is a quotient of $\langle t \rangle$ by a congruence, and the monoid $G \cup \{ 0\}$ for a finite group $G$.