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.

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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.

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We associate to a projective n-dimensional toric variety X∆ a pair of cocommutative (but generally non-commutative) Hopf algebras H α X , H T X . These arise as Hall algebras of certain categories Coh α (X), Coh T (X) of coherent sheaves on X∆ viewed as a monoid scheme -i.e. a scheme obtained by gluing together spectra of commutative monoids rather than rings. When X is smooth, the category Coh T (X) has an explicit combinatorial description as sheaves whose restriction to each A n corresponding to a maximal cone is determined by an n-dimensional generalized skew shape. The (non-additive) categories Coh α (X), Coh T (X) are treated via the formalism of protoexact/proto-abelian categories developed by Dyckerhoff-Kapranov.The Hall algebras H α X , H T X are graded and connected, and so enveloping algebras, where the Lie algebras n α X , n T X are spanned by the indecomposable coherent sheaves in their respective categories.We explicitly work out several examples, and in some cases are able to relate n T X to known Lie algebras. In particular, whenWe also consider the case X = P 2 , where we give a basis for n T X by describing all indecomposable sheaves in Coh T (X).

Recently, M. Baker and N. Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields. Date: December 27, 2017. 2010 Mathematics Subject Classification. 05E99(primary), 16T05(secondary). Key words and phrases. matroid, hyperfield, matroid over hyperfields, Hopf algebra, minor, direct sum.In this sense B and C B carry the same information as C B and B C respectively; these are thus said to determine the same matroid on E "cryptomorphically."Example 2.2. The motivating examples of matroids (hinted at above) are given as follows:(1) Let V be a finite dimensional vector space and E ⊆ V a spanning set of vectors.The bases of V contained in E form the bases of a matroid on E, and the minimal dependent subsets of E form the circuits of a matroid on E. Furthermore, these are the same matroid.(2) Let Γ be a finite, undirected graph with edge set E (loops and parallel edges are allowed). The sets of edges of spanning forests in Γ form the bases of a matroid on E, and the sets of edges of cycles form the circuits of a matroid on E. Furthermore, these are the same matroid (called the graphic matroid of Γ).One can define the notion of isomorphisms of matroids as follows.Definition 2.3. Let M 1 (resp. M 2 ) be a matroid on E 1 (resp. E 2 ) defined by a set B 1 (resp. B 2 ) of bases. We say that M 1 is isomorphic to M 2 if there exists a bijection f :In this case, f is said to be an isomorphism.Example 2.4. Let Γ 1 and Γ 2 be finite graphs and M 1 and M 2 be the corresponding graphic matroids. Every graph isomorphism between Γ 1 and Γ 2 gives rise to a matroid isomorphism between M 1 and M 2 , but the converse need not hold.Recall that given any base B ∈ B(M ) and any element e ∈ E \ B, there is a unique circuit (fundamental circuit ) C B,e of e with respect to B such that C B,e ⊆ B ∪ {e}.One can construct new matroids from given matroids as follows:Definition 2.5 (Direct sum of matroids). Let M 1 and M 2 be matroids on E 1 and E 2 given by bases B 1 and B 2 respectively. The direct sum M 1 ⊕ M 2 is the matroid on E 1 ⊔ E 2 given by the bases B = {B 1 ⊔ B 2 | B i ∈ B i for i = 1, 2}.Remark 2.6. One can easily check that M 1 ⊕ M 2 is indeed a matroid on E 1 ⊔ E 2 .Definition 2.7 (Dual, Restriction, Deletion, and Contraction). Let M be a matroid on a finite set E M with the set B M of bases and the set C M of circuits. Let S be a subset of E M .(1) The dual M * of M is a matroid on E M given by bases

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