On the Hall algebra of semigroup representations over $$\mathbb F _1$$ F 1

Abstract

Let A be a finitely generated semigroup with 0. An A-module over F 1 (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 Hall Lie algebra of C A . In the case of the t -the free monoid on one generator t , the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent t -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 t by a congruence, and the monoid G ∪ {0} for a finite group G.

The Hopf algebra of skew shapes, torsion sheaves on A/F1n, and ideals in Hall algebras of monoid

“…The following simple result is proved in [29]: Proposition 2.1.1. Let A be a monoid and A-mod n as above.…”

“…H A is spanned by δ-functions δ M ∈ H A supported on individual isomorphism classes, and so it is useful to make explicit the multiplication of two such elements. We have The following theorem is proved in [29]: 0.2. (H A , , ∆) is a graded, connected, co-commutative bialgebra.…”

“…It is clear that every irreducible module is indecomposable. We have the following analogue of the Krull-Schmidt theorem ( [29]): Proposition 2.0.1. Every M ∈ A-mod can be uniquely decomposed (up to reordering) as a direct sum of indecomposable A-modules.…”

“…An A-module M may be described in terms of a directed graph with an edge from m to t · m if t · m 0. The possible directed graphs arising this way from indecomposable A-modules were classified in [29]. These are either rooted trees (Figure 1), or a cycle with rooted trees attached ( Figure 2).…”

“…As shown in [28,29] the corresponding Hall algebras can be described as follows (1) We have that H 0,A = H gr A = H gr 0,A . This Hopf algebra is dual to the Connes-Kreimer Hopf algebra of rooted forests.…”