A non-simply laced version for cluster structures on 2-Calabi-Yau categories

This paper investigates a non simply-laced version of cluster structures for 2-Calabi-Yau or stably 2-Calabi-Yau categories over arbitrary fields. It results that 2-Calabi-Yau or stably 2-Calabi-Yau categories having a cluster tilting subcategory with neither loops nor 2-cycles do have the generalized version of cluster structure. This is in particular the case of cluster categories over non-algebraically closed fields.Comment: Beamer version. Journal of Pure and Applied Algebra, Available online 1 December 2013 http://dx.doi.org/10.1016/j.jpaa.2013.11.02

Existence of random gradient states

We consider two versions of random gradient models. In model A the interface feels a bulk term of random fields while in model B the disorder enters through the potential acting on the gradients. It is well known that for gradient models without disorder there are no Gibbs measures in infinite-volume in dimension d=2, while there are "gradient Gibbs measures" describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn. Van Enter and K\"{u}lske proved that adding a disorder term as in model A prohibits the existence of such gradient Gibbs measures for general interaction potentials in $d=2$. In the present paper we prove the existence of shift-covariant gradient Gibbs measures with a given tilt $u\in \mathbb{R}^d$ for model A when $d\geq3$ and the disorder has mean zero, and for model B when $d\geq1$. When the disorder has nonzero mean in model A, there are no shift-covariant gradient Gibbs measures for $d\ge3$. We also prove similar results of existence/nonexistence of the surface tension for the two models and give the characteristic properties of the respective surface tensions.Comment: Published in at http://dx.doi.org/10.1214/11-AAP808 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The double Ringel-Hall algebra on a hereditary abelian finitary length category

In this paper, we study the category $\mathscr{H}^{(\rho)}$ of semi-stable coherent sheaves of a fixed slope $\rho$ over a weighted projective curve. This category has nice properties: it is a hereditary abelian finitary length category. We will define the Ringel-Hall algebra of $\mathscr{H}^{(\rho)}$ and relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type theorem to describe the indecomposable objects in this category, i.e. the indecomposable semi-stable sheaves.Comment: 29 page

The preprojective algebra of a modulated graph

Dlab V, Ringel CM. The preprojective algebra of a modulated graph. In: Dlab V, Gabriel P, eds. Representation Theory II. Proceedings of the Second International Conference on Representations of Algebras, Ottawa, Carleton University, August 13-25, 1979: No. 2. Lecture Notes in Mathematics. Vol 832. Berlin, Heidelberg: Springer; 1980: 216-231

Auslander algebras as quasi-hereditary algebras

Dlab V, Ringel CM. Auslander algebras as quasi-hereditary algebras. Journal of the London Mathematical Society : Ser. 2. 1989;39(3):457-466

A-D-E Quivers and Baryonic Operators

We study baryonic operators of the gauge theory on multiple D3-branes at the tip of the conifold orbifolded by a discrete subgroup Gamma of SU(2). The string theory analysis predicts that the number and the order of the fixed points of Gamma acting on S^2 are directly reflected in the spectrum of baryonic operators on the corresponding quiver gauge theory constructed from two Dynkin diagrams of the corresponding type. We confirm the prediction by developing techniques to enumerate baryonic operators of the quiver gauge theory which includes the gauge groups with different ranks. We also find that the Seiberg dualities act on the baryonic operators in a non-Abelian fashion.Comment: 46 pages, 17 figures; v2: minor corrections, note added in section 1, references adde

Recollements of Module Categories

We establish a correspondence between recollements of abelian categories up to equivalence and certain TTF-triples. For a module category we show, moreover, a correspondence with idempotent ideals, recovering a theorem of Jans. Furthermore, we show that a recollement whose terms are module categories is equivalent to one induced by an idempotent element, thus answering a question by Kuhn.Comment: Comments are welcom