On the Hall algebra of semigroup representations over F_1

Abstract

Let $\A$ be a finitely generated semigroup with 0. An $\A$-module over $\fun$ (also called an $\A$--set), is a pointed set $(M,*)$ together with an action of $\A$. We define and study the Hall algebra $\H_{\A}$ of the category $\C_{\A}$ of finite $\A$--modules. $\H_{\A}$ is shown to be the universal enveloping algebra of a Lie algebra $\n_{\A}$, called the \emph{Hall Lie algebra} of $\C_{\A}$. In the case of the $\fm$ - the free monoid on one generator $\fm$, the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\fm$-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 $\A$ is a quotient of $\fm$ by a congruence, and the monoid $G \cup \{0\}$ for a finite group $G$.