On the Hall algebra of semigroup representations over \(\mathbb F_1\). (Q2636957)

This paper introduces and studies the Hall algebra \(\mathbb H_A\) of the category of representations of a semigroup \(A\) over \(\mathbb F_1\), that is to say in the category of pointed sets. With a restriction of normality on the morphisms, such a Hall algebra exists, and is the enveloping algebra of a certain Lie algebra. When \(A\) is the free monoid \(\langle t\rangle\) on one generator, the elements of \(\mathbb H_A\) are represented by oriented graphs, and submodules correspond to admissible cuts. Restricting to nilpotent representations, the corresponding graphs are rooted forests and we obtain the Connes-Kreimer Hopf algebra of rooted trees. Hall algebras of quotients of \(\langle t\rangle\) by a congruence and of monoids \(G\sqcup\{0\}\), where \(G\) is a group, are also described.