ABSTRACT. We define and study the category Rep(Q, F 1 ) of representations of a quiver in Vect(F 1 ) -the category of vector spaces "over F 1 ". Rep(Q, F 1 ) is an F 1 -linear category possessing kernels, co-kernels, and direct sums. Moreover, Rep(Q, F 1 ) satisfies analogues of the Jordan-Hölder and Krull-Schmidt theorems. We are thus able to define the Hall algebra H Q of Rep(Q, F 1 ), which behaves in some ways like the specialization at q = 1 of the Hall algebra of Rep(Q, F q ). We prove the existence of a Hopf algebra homomorphism of ρ : U(n + ) → H Q , from the enveloping algebra of the nilpotent part n + of the Kac-Moody algebra with Dynkin diagram Q -the underlying unoriented graph of Q. We study ρ when Q is the Jordan quiver, a quiver of type A, the cyclic quiver, and a tree respectively.

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.

No abstract

a b s t r a c tGiven a family F of posets closed under disjoint unions and the operation of taking convex subposets, we construct a category C F called the incidence category of F . This category is ''nearly abelian'' in the sense that all morphisms have kernels/cokernels, and possesses a symmetric monoidal structure akin to direct sum. The Ringel-Hall algebra of C F is isomorphic to the incidence Hopf algebra of the collection P (F ) of order ideals of posets in F . This construction generalizes the categories introduced by K. Kremnizer and the author, in the case when F is the collection of posets coming from rooted forests or Feynman graphs.

No abstract