Let A be the path algebra of a Euclidean quiver over a finite field k. The aim of this paper is to classify the modules M with the property [M ] ∈ C(A), where C(A) is Ringel’s composition algebra. Namely, the main result says that if |k| 6= 2, 3, then [M ] ∈ C(A) if and only if the regular direct summand of M is a direct sum of modules from non-homogeneous tubes with quasi-dimension vectors non...

We compute the quiver of any finite monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as DO) to representation-theoretic computations for group algebras of maximal sub...

A graph monoid is a commutative monoid for which there is a particularly simple presentation, given in terms of a quiver. Such monoids are known to satisfy various nonstable K-theoretical representability properties for either von Neumann regular rings or C*-algebras. We give a characterization of graph monoids within finitely generated antisymmetric refinement monoids. This characterization is...

In this article, we state the necessary background for cluster algebras and quiver representations to formulate and prove a result of Caldero and Chapoton which gives a nice formula connecting the two subjects. In our exposition, we mostly follow the paper [CC], but we have tried to minimize the reliance of citing outside sources as much as possible. In particular, we avoid having to introduce ...

We situate the noncrossing partitions associated to a finite Coxeter group within the context of the representation theory of quivers. We describe Reading’s bijection between noncrossing partitions and clusters in this context, and show that it extends to the extended Dynkin case. Our setup also yields a new proof that the noncrossing partitions associated to a finite Coxeter group form a latti...

For a finite abelian group G ⊂ GL(n, k), we describe the coherent component Yθ of the moduli space Mθ of McKay quiver representations. This is a not-necessarily-normal toric variety that admits a projective birational morphism Yθ → A/G obtained by variation of GIT quotient. We present a simple calculation to determine the quiver representation corresponding to any point of Yθ, and describe the ...

In this article, we give geometric constructions of tensor products in various categories using quiver varieties. More precisely, we introduce a lagrangian subvariety Z̃ in a quiver variety, and show the following results: (1) The homology group of Z̃ is a representation of a symmetric Kac-Moody Lie algebra g, isomorphic to the tensor product V (λ1)⊗· · ·⊗V (λN ) of integrable highest weight modu...

There are three types of monopole in gauge theories with fundamental matter and N = 2 supersymmetry broken by a superpotential. There are unconfined 0-monopoles and also 1 and 2-monopoles confined respectively by one or two vortices transforming under distinct components of the unbroken gauge group. If a Fayet-Iliopoulos term is added then there are only 2-monopoles. Monopoles transform in the ...

Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated modules over their centers and their representation spaces are toric varieties. In order to describe these algebras we introduce the notion of a toric order and s...

We prove there is an equivalence of derived categories between Orlov’s triangulated category of singularities for a Gorenstein cyclic quotient singularity and the derived category of representations of a quiver with relations, which is obtained from a McKay quiver by removing one vertex and half of the arrows. This result produces examples of distinct quivers with relations which have equivalen...