Morphism spaces in stable categories of Frobenius algebras

We present an explicit formula computing morphism spaces in the stable category of a Frobenius algebra.     

Morita classes of algebras in modular tensor categories

We consider algebras in a modular tensor category C. If the trace pairing of an algebra A in C is non-degenerate we associate to A a commutative algebra Z(A), called the full centre, in a doubled version of the category C. We prove that two simple algebras with non-degenerate trace pairing are Morita-equivalent if and only if their full centres are isomorphic as algebras. This result has an interesting interpretation in two-dimensional rational conformal field theory; it implies that there cannot be several incompatible sets of boundary conditions for a given bulk theory.     

Nilpotent fusion categories

In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series of C. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger.     

From triangulated categories to abelian categories: cluster tilting in a general framework

A general framework for cluster tilting is set up by showing that any quotient of a triangulated category modulo a tilting subcategory (i.e., a maximal 1-orthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.      

Koszulity of some path categories

We prove Koszulity of certain linear path categories obtained from connected graphs with some infinite directed walk. These categories can be viewed as locally quadratic dual to preprojective algebras.     

Coloured quiver mutation for higher cluster categories

We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting in higher cluster categories and especially an algorithm to determine the Gabriel quivers of tilting objects in such categories.     

Weakly group-theoretical and solvable fusion categories

We introduce two new classes of fusion categories which are obtained by a certain procedure from finite groups – weakly group-theoretical categories and solvable categories. These are fusion categories that are Morita equivalent to iterated extensions (in the world of fusion categories) of arbitrary, respectively solvable finite groups. Weakly group-theoretical categories have integer dimension, and all known fusion categories of integer dimension are weakly group-theoretical. Our main results are that a weakly group-theoretical category C has the strong Frobenius property (i.e., the dimension of any simple object in an indecomposable C-module category divides the dimension of C), and that any fusion category whose dimension has at most two prime divisors is solvable (a categorical analog of Burnside's theorem for finite groups). This has powerful applications to classification of fusion categories and semsisimple Hopf algebras of a given dimension. In particular, we show that any fusion category of integer dimension <84 is weakly group-theoretical (i.e. comes from finite group theory), and give a full classification of semisimple Hopf algebras of dimensions pqr and pq², where p,q,r are distinct primes.     

BGP reflection functors in root categories

We define the BGP-reflection functors in the derived categories and the root categories. By Ringel's Hall algebra approach, the BGP-reflection functor is applicable to obtain the classical Weyl group action on the Lie algebra.     

A recollement construction of Gorenstein derived categories

We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause’s work [Math. Ann., 2012, 353: 765–781]. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived category of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.     

Coherence for closed categories with biproducts

A coherence result for symmetric monoidal closed categories with biproducts is shown in this paper. It is also explained how to prove coherence for compact closed categories with biproducts and for dagger compact closed categories with dagger biproducts by using the same technique.     

Gorenstein global dimension of recollements of abelian categories

Abstract

In this article, we investigate the Gorenstein global dimension with respect to the recollements of abelian categories. With the invariants spli and silp of the categories, we give some upper bounds of Gorenstein global dimensions of the categories involved in a recollement of abelian categories. We apply our results to some rings and artin algebras, especially to the triangular matrix artin algebras.     

最大k—一致超图

本文刻划直径为ｄ 的最大边数的ｋ－一致超图的结构,推广了Ｏｒｅ 的一个结果     

Hypergraph decomposition and secret sharing

A secret sharing scheme is a protocol by which a dealer distributes a secret among a set of participants in such a way that only qualified sets of them can reconstruct the value of the secret whereas any non-qualified subset of participants obtain no information at all about the value of the secret. Secret sharing schemes have always played a very important role for cryptographic applications and in the construction of higher level cryptographic primitives and protocols.In this paper we investigate the construction of efficient secret sharing schemes by using a technique called hypergraph decomposition, extending in a non-trivial way the previously studied graph decomposition techniques. A major advantage of hypergraph decomposition is that it applies to any access structure, rather than only structures representable as graphs. As a consequence, the application of this technique allows us to obtain secret sharing schemes for several classes of access structures (such as hyperpaths, hypercycles, hyperstars and acyclic hypergraphs) with improved efficiency over previous results. Specifically, for these access structures, we present secret sharing schemes that achieve optimal information rate. Moreover, with respect to the average information rate, our schemes improve on previously known ones.In the course of the formulation of the hypergraph decomposition technique, we also obtain an elementary characterization of the ideal access structures among the hyperstars, which is of independent interest.     

The ubiquity of generalized cluster categories

Associated with a finite-dimensional algebra of global dimension at most 2, a generalized cluster category was introduced in Amiot (2009) [1]. It was shown to be triangulated, and 2-Calabi–Yau when it is Hom-finite. By definition, the cluster categories of Buan et al. (2006) [4] are a special case. In this paper we show that a large class of 2-Calabi–Yau triangulated categories, including those associated with elements in Coxeter groups from Buan et al. (2009) [7], are triangle equivalent to generalized cluster categories. This was already shown for some special elements in Amiot (2009) [1].     

Decidable (= separable) objects and morphisms in lextensive categories

After investigating all conceivable properties of decidable objects and maps in left exact categories with well-behaved finite sums (‘lextensive categories’), we give a characterization in such categories of decidable morphisms which are (finite) coverings (in an appropriate sense). Finally, we give two applications of this result, to separable algebras and to local homeomorphisms. In both cases it explains categorically the advantage of two well-known notions — strongly separable algebras and local homeomorphisms with path lifting property, respectively.     

Implicit operations on the categories of universal algebras

We present a series of concepts and prove some results on implicit operations on the various categories of universal algebras. This generalizes the previous results for pseudovarieties, pseudouniversal classes of algebras, positively conditional pseudovarieties, and so on.     

Yetter—Drinfel'd H-Azumaya Monoids in Closed Categories

When C is a symmetric closed category with equalizers and coequalizers and H is a Hopf algebra in C, the category of Yetter—Drinfeld H-modules is a braided monoidal category.We develop a categorical version of the results in (10) constructing a Brauer group BQ(C,H) and studying its functorial properties.     

Models for singularity categories

In this article we construct various models for singularity categories of modules over differential graded rings. The main technique is the connection between abelian model structures, cotorsion pairs and deconstructible classes, and our constructions are based on more general results about localization and transfer of abelian model structures. We indicate how recollements of triangulated categories can be obtained model categorically, discussing in detail Krause?s recollement _Kac(Inj(R))→K(Inj(R))→D(R)${\mathrm{K}}_{\mathrm{ac}}(\mathrm{Inj}(R))\to \mathrm{K}(\mathrm{Inj}(R))\to \mathrm{D}(R)$. In the special case of curved mixed Z$\mathbb{Z}$-graded complexes, we show that one of our singular models is Quillen equivalent to Positselski?s contraderived model for the homotopy category of matrix factorizations.     

Intertwining operator superalgebras and vertex tensor categories for superconformal algebras, II

We construct the intertwining operator superalgebras and vertex tensor categories for the superconformal unitary minimal models and other related models.