In this paper, we prove that Graded Calabi Yau algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau algebras, is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist.

Abstract. Dimer models are a combinatorial tool to describe certain algebras that appear as noncommutative crepant resolutions of toric Gorenstein singularities. Unfortunately, not every dimer model gives rise to a noncommutative crepant resolution. Several notions of consistency have been introduced to deal with this problem. In this paper, we study the major different notions in detail and show that for dimer models on a torus, they are all equivalent.2010 Mathematics Subject Classification. 14M25, 14A22, 16S38. Introduction.If X is a three-dimensional normal Gorenstein singularity admitting a crepant resolutionX → X, then one is interested to describe the bounded derived category DCohX of coherent sheaves onX. A well-known result by Bridgeland [4] shows that this category only depends on the singularity and not on the particular choice of crepant resolution.In many cases, there exists a tilting bundle in U ∈ DCohX such that DCohX is equivalent as a triangulated category to the derived category of finitely generated A-modules DModA, where A = EndX . To model these algebras without referring to a commutative crepant resolution, Van den Bergh [23] introduced the notion of a noncommutative crepant resolution (NCCR) of X. This is a homologically homogeneous algebra of the form A = End R (T), where T is a reflexive R-module, with R = ‫[ރ‬X] the coordinate ring of the singularity. An NCCR is, however, far from unique and in general there are an infinite number of different noncommutative crepant resolutions.If we make two restrictions, the problem becomes more manageable. First, we assume that X is a toric three-dimensional Gorenstein singularity. This automaticly implies the existence of a commutative crepant resolution. Secondly, we assume that the tilting bundle is a direct sum of nonisomorphic line bundles. It was noticed in string theory [8,10,11] that, under these conditions, the algebra A can be described using a dimer model on a torus.This means that A is the path algebra of a quiver Q with relations where Q is embedded in a two-dimensional real torus T such that every connected piece of T \ Q is bounded by a cyclic path of length at least 3. The relations are given by demanding for every arrow a that p = q where ap and aq are the bounding cycles that contain a.This nice description follows from the fact that the algebra A is a toric order, a special type of order compatible with the toric structure, and Calabi-Yau-3 (CY-3), i.e. it admits a self-dual bimodule resolution of length 3. In [2], it was shown that every toric CY-3 order comes from a dimer model. Not every dimer model gives a noncommutative crepant resolution of its centre. To do so, it needs to satisfy some extra conditions called consistency conditions. In recent years,

We present an algorithm that finds all toric noncommutative crepant resolutions of a given toric 3-dimensional Gorenstein singularity. The algorithm embeds the quivers of these algebras inside a real 3-dimensional torus such that the relations are homotopy relations. One can project these embedded quivers down to a 2-dimensional torus to obtain the corresponding dimer models. We discuss some examples and use the algorithm to show that all toric noncommutative crepant resolutions of a finite quotient of the conifold singularity can be obtained by mutating one basic dimer model. We also discuss how this algorithm might be extended to higher dimensional singularities.

Multiple access algorithms without feedback using combinatorial designs IEEE transactions on communications / IEEE [New York, N.Y.] -

We study the Luna slice theorem in the case of quivers with an involution or supermixed quivers as introduced by Zubkov in [6]. We construct an analogue to the notion of a local quiver setting described in [3]. We use this technique to determine dimension vectors of simple supermixed representations.