Cluster structures for 2-Calabi-Yau categories and unipotent groups

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related to unipotent groups, both in the Dynkin and non Dynkin case.Comment: 49 pages. For the third version the presentation is revised, especially Chapter III replaces the old Chapter III and I

Cluster algebras of type A2(1)A_2^{(1)}

In this paper we study cluster algebras \myAA of type $A_2^{(1)}$. We solve the recurrence relations among the cluster variables (which form a T--system of type $A_2^{(1)}$). We solve the recurrence relations among the coefficients of \myAA (which form a Y--system of type $A_2^{(1)}$). In \myAA there is a natural notion of positivity. We find linear bases \BB of \myAA such that positive linear combinations of elements of \BB coincide with the cone of positive elements. We call these bases \emph{atomic bases} of \myAA. These are the analogue of the "canonical bases" found by Sherman and Zelevinsky in type $A_{1}^{(1)}$. Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of \BB are parameterized by \ZZ^3 via their $\mathbf{g}$--vectors in every cluster. We prove that the denominator vector map in every acyclic seed of \myAA restricts to a bijection between \BB and \ZZ^3. In particular this gives an explicit algorithm to determine the "virtual" canonical decomposition of every element of the root lattice of type $A_2^{(1)}$. We find explicit recurrence relations to express every element of \myAA as linear combinations of elements of \BB.Comment: Latex, 40 pages; Published online in Algebras and Representation Theory, springer, 201

The first Hochschild cohomology group of a schurian cluster-tilted algebra

Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH1(B) with coefficients in the B-B-bimodule B. We find several consequences when B is representation-finite, and also in the case where B is cluster-tilted of type Ã.Fil: Assem, Ibrahim. University of Sherbrooke; CanadáFil: Redondo, Maria Julia. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Matemática Bahía Blanca. Universidad Nacional del Sur. Departamento de Matemática. Instituto de Matemática Bahía Blanca; Argentin

Torsion pairs and rigid objects in tubes

We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Pruefer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs, maximal rigid objects and the bijection between them geometrically. We also give a similar geometric description in the case of the linear orientation of a Dynkin quiver of type A.Comment: 25 pages, 13 figures. Paper shortened. Minor errors correcte

Repetitive higher cluster categories of type A_n

We show that the repetitive higher cluster category of type A_n, defined as the orbit category D^b(mod kA_n)/(tau^{-1}[m])^p, is equivalent to a category defined on a subset of diagonals in a regular p(nm+1)-gon. This generalizes the construction of Caldero-Chapoton-Schiffler, which we recover when p=m=1, and the work of Baur-Marsh, treating the case p=1, m>1. Our approach also leads to a geometric model of the bounded derived category D^b(mod kA_n)

MORFOLOGI KAWASAN KOTA LAMA KUPANG

Kota Kupang Merupakan ibu kota provinsi Nusa Tenggara Timur dan menjadi kota terbesar di pulau Timor. Kota Kupang terletak di pesisir teluk Kupang, bagian barat laut pulau Timor. Menurut sejarah terbentuknya kota Kupang berawal dari kota bandar yang dikuasai oleh Raja Helong yaitu kawasan kota lama Kupang. Perkembangan kawasan kota lama Kupang dimulai pada periode abad ke 15. Dalam perkembangan kawasan kota lama Kupang terdapat faktor-faktor yang mempengaruhi penting yang mempengaruhi morfologi kawasan kota lama yang terjadi selama beberapa periode. Tujuan dari penulisan ini adalah memberikan gambaran mengenai perkembangan kawasan kota lama Kupang selama beberapa periode dan melihat perubahan dan perbandingan apa saja terkait morfologi kawasan beserta faktor-faktor yang mempengaruhi morfologi kawasan kota lama kupang. Penelitian ini merupakan penelitian deskriptif eksploratif dengan metode analisis sinkronik (tissue analysis) digunakan untuk membaca sejarah yang terjadi pada kawasan kota lama Kupang secara beberapa periode waktu atau pada abad ke 15 awal terbentuk sampai pada abad ke 21 dan Diakronik (historical reading) digunakan melihat perubahan dan perbandingan Morfologi kawasan kota lama Kupang periode abad ke 15 sampai 21 dan memaparkan bagiamana ruang-ruang tersebut mulai bertumbuh, berkembang dari hasil analisis tersebut akan didapatkan faktor-faktor yang mempengaruhi morfologi kawasan kota lama Kupang dengan menggunakan teori Branch (1995). Studi ini menghasilkan kesimpulan bahwa kawasan kota lama Kupang mulai berkembang karena memiliki generator utama ialah masuknya Raja Helong dan kawasan ini menjadi salah satu kota bandar yang ada di pulau Timor. Kemudian Perubahan dan perbandingan perkembangan morfologi kawasan kota lama Kupang saat masuknya Bangsa Belanda,Portugis dan Etnis Cina dan saat ditetapkan batas-batas kota pada tahun 1886 serta kebijakan-kebijakan politik terhadap status kawasan setelah Indonesia merdeka. faktor-faktor yang mempengaruhi morfologi kawasan kota lama Kupang berupa faktor Sejarah dan Budaya, Geografis, politik, Sosial dan Ekonomi

Generalised Moore spectra in a triangulated category

In this paper we consider a construction in an arbitrary triangulated category T which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of T satisfying some finite tilting assumptions, we obtain a functor which "approximates" objects of the module category of the endomorphism algebra of C in T. This generalises and extends a construction of Jorgensen in connection with lifts of certain homological functors of derived categories. We show that this new functor is well-behaved with respect to short exact sequences and distinguished triangles, and as a consequence we obtain a new way of embedding the module category in a triangulated category. As an example of the theory, we recover Keller's canonical embedding of the module category of a path algebra of a quiver with no oriented cycles into its u-cluster category for u>1.Comment: 26 pages, improvement to exposition of the proof of Theorem 3.

Cluster algebras in algebraic Lie theory

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.Comment: Invited survey; to appear in Transformation Group

Comment on "Is the nonlinear Meissner effect unobservable?"

In a recent Letter (Phys. Rev. Lett. 81, p.5640 (1998), cond-mat/9808249 v3), it was suggested that nonlocal effects may prevent observation of the nonlinear Meissner effect in YBCO. We argue that this claim is incorrect with regards to measurements of the nonlinear transverse magnetic moment, and that the most likely reason for a null result lies elsewhere.Comment: 1 pag