Representations of Quivers Over F1 and Hall Algebras

Abstract

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.

Maximum antichains in posets of quiver representations

“…A representation of Q with values in the category of pointed sets is a collection X = ((X i , 0 X i ) i∈Q 0 , (X α ) α∈Q 1 ) consisting of a pointed set (X i , 0 X i ) for every vertex i ∈ Q 0 and a morphism of pointed sets Szczesny (2012) studies such representations. Let us recall some basic constructions.…”

“…Szczesny (2012), Theorem 5 describes indecomposable representations with values in the category of pointed sets for quivers whose underlying undirected diagram is a tree. In this case, the isomorphism classes of indecomposable representations correspond to connected subquivers Q of Q.…”

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

“…Here, Ob(C) typically consist of combinatorial structures equipped an operation of "collapsing" a sub-structure, which corresponds to forming a quotient in C. Examples of such C include trees, graphs, posets, matroids, semigroup representations on pointed sets, quiver representations in pointed sets etc. (see [18,[26][27][28][29] ). The product in H C , which counts all extensions between two objects, thus amounts to enumerating all combinatorial structures that can be assembled from the two.…”

“…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.…”

Relative 2–Segal spaces

“…The input for the R • -construction is a proto-exact category with duality which satisfies a reduction assumption. In the case of exact categories the R • -construction categorifies the Hall algebra representations of [43], [10], [46], [47] while for the proto-exact category Rep F1 (Q) of representations of a quiver over F 1 we obtain new modules over Szczesny's combinatorial Hall algebras [41]. The latter modules will be the subject of future work.…”