On the Structure of Modular Categories

For a braided tensor category C and a subcategory K there isa notion of a centralizer C_C K, which is a full tensor subcategoryof C. A pre-modular tensor category is known to be modular inthe sense of Turaev if and only if the center Z₂C C_CC (not tobe confused with the center Z₁ of a tensor category, relatedto the quantum double) is trivial, that is, consists only ofmultiples of the tensor unit, and dimC 0. Here , the X_i being the simple objects. We prove several structural properties of modular categories.Our main technical tool is the following double centralizertheorem. Let C be a modular category and K a full tensor subcategoryclosed with respect to direct sums, subobjects and duals. ThenC_CC_CK = K and dim K·dim C_CK = dim C. We give several applications. (1) If C is modular and K is a full modular subcategory,then L=C_CK is also modular and C is equivalent as a ribbon categoryto the direct product: . Thus every modular category factorizes (non-uniquely, in general)into prime modular categories. We study the prime factorizationsof the categories D(G)-Mod, where G is a finite abelian group. (2) If C is a modular *-category and K is a full tensorsubcategory then dim C dim K · dim Z₂K. We give exampleswhere the bound is attained and conjecture that every pre-modularK can be embedded fully into a modular category C with dim C=dimK·dim Z₂K. (3) For every finite group G there is a braided tensor*-category C such that Z₂CRep,G and the modular closure/modularization is non-trivial. 2000 MathematicsSubject Classification 18D10.     

Square-Free Non-Cayley Numbers. On Vertex-Transitive Non-Cayley Graphs of Square-Free Order

A complete classification is given of finite primitive permutation groups which contain a regular subgroup of square-free order. Then a collection of square-free numbers n is obtained such that there exists a vertex-primitive non-Cayley graph on n vertices if and only if n is a member of .     

Holomorphic Almost Modular Forms

Holomorphic almost modular forms are holomorphic functions ofthe complex upper half plane that can be approximated arbitrarilywell (in a suitable sense) by modular forms of congruence subgroupsof large index in SL(2,Z). It is proved that such functionshave a rotation-invariant limit distribution when the argumentapproaches the real axis. An example of a holomorphic almostmodular form is the logarithm of . The paper is motivated by the author's previous studies [Int.Math. Res. Not. 39 (2003) 2131–2151] on the connectionbetween almost modular functions and the distribution of thesequence n²x modulo one. 2000 Mathematics Subject Classification11F11 (primary), 11F06, 11J71 (secondary).     

On the Exponential Sum with Square-Free Numbers

A new estimate for the exponential sum with square-free numbersis established. This result is applied to the problem of findingthe number of representations of a large integer as a sum ofthree square-free numbers. 2000 Mathematics Subject Classification11L07, 11N36, 11P99.     

Classification of Abelian Track Categories

We show that each category enriched in Abelian groupoids is a linear track extension and hence is determined up to weak equivalence by a characteristic chomology class. We also discuss compatibility with coproducts.     

Multipliers of a Family of Almost Modular Functions

The goal of this paper is to complete an investigation begun by Cohn and Knopp in their 1994 paper, Application of Dedekind eta-multipliers to modular equations. The paper concerned _k (z), a family of modular forms on ⁰(N) (N a positive integer) with possibly non-trivial multiplier systems. Cohn and Knopp defined new functions _k (z) and a new group containing ⁰(N) and proved that for all S in the larger group and for all k, _k (Sz) = M _k(S) _k (z), where M _k(S)²⁴ = 1. This yielded interesting invariance properties of _k , dependent on the values of M _k(S). Fixing a constant integer e, independent of k, Cohn and Knopp proved that for all k and all S in the larger group, M _k(S) ^e = (±1) ^e . They determined the sign of M _k(S) ^e in many, but not all, cases. In this paper, we give a complete determination of the values of M _k(S) ^e in the remaining cases.     

Galois Theory for Braided Tensor Categories and the Modular Closure

Given a braided tensor *-category with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory we define a crossed product . This construction yields a tensor *-category with conjugates and an irreducible unit. (A *-category is a category enriched over Vect with positive *-operation.) A Galois correspondence is established between intermediate categories sitting between and and closed subgroups of the Galois group Gal( / )=Aut ( ) of , the latter being isomorphic to the compact group associated with by the duality theorem of Doplicher and Roberts. Denoting by the full subcategory of degenerate objects, i.e., objects which have trivial monodromy with all objects of , the braiding of extends to a braiding of iff . Under this condition, has no non-trivial degenerate objects iff = . If the original category is rational (i.e., has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category ≡ is called the modular closure of since in the rational case it is modular, i.e., gives rise to a unitary representation of the modular group SL(2,  ). If all simple objects of have dimension one the structure of the category can be clarified quite explicitly in terms of group cohomology.     

S-Structures for k-Linear Categories and the Definition of a Modular Functor

Ideas from string theory and quantum field theory have beenthe motivation for new invariants of knots and 3-dimensionalmanifolds which have been constructed from complex algebraicstructures such as Hopf algebras [17, 22], monoidal categorieswith additional structure [24], and modular functors [14, 23].These constructions are closely related. Here we take a unifyingcategorical approach based on a natural 2-dimensional generalisationof a topological field theory in the sense of Atiyah [1], andshow that the axioms defining these complex algebraic structuresare a consequence of the underlying geometry of surfaces.     

Modular Group Algebras with Almost Maximal Lie Nilpotency Indices

Let be a field of positive characteristic and the group algebra of a group . It is known that, if is Lie nilpotent, then its upper (and lower) Lie nilpotency index is at most , where is the order of the commutator subgroup. The authors previously determined those groups for which this index is maximal and here they determine the groups for which it is `almost maximal', that is, it takes the next highest possible value, namely .Presented by V. Dl a b.Dedicated to Professor Vjacheslav Rudko on his 65th birthday.The research was supported by OTKA No. T 037202, No. T 038059 and Italian National Research Project “Group Theory and Application.”     

Classifying Arc-Transitive Circulants of Square-Free Order

A circulant is a Cayley graph of a cyclic group. Arc-transitive circulants of square-free order are classified. It is shown that an arc-transitive circulant of square-free order n is one of the following: the lexicographic product , or the deleted lexicographic , where n = bm and is an arc-transitive circulant, or is a normal circulant, that is, Aut has a normal regular cyclic subgroup.     

Arc-Transitive Pentavalent Graphs of Square-Free Order

Hua et al. (Discrete Math 311, 2259–2267, 2011) and Yang et al. (Discrete Math. 339, 522–532, 2016) classify arc-transitive pentavalent graphs of order 2pq and of order 2pqr (with p, q, r distinct odd primes), respectively. In this paper, we extend their results by giving a classification of arc-transitive pentavalent graphs of any square-free order.     

On Almost Interpolation

We obtain some characterizations of almost interpolation configurations of points with respect to finite-dimensional functional spaces. Particularly, a Schoenberg–Whitney type characterization which is valid for any multivariate spline space relative to an arbitrary partition of a domainA⊂^mis presented. As a closely related problem we investigate sectional structure of finite-dimensional spaces of real functions on a topological spaceA. It is shown that under some reasonable restrictions onAany space of this sort may be considered as piecewise almost Chebyshev.     

范畴的K—根与Jacobson结构定理

本文得到三方面结果：（1）定义加法范畴的K－根，给出它的模刻划式。（2）给出J－根的内部刻划。（3）给出J－半单范畴结构中由本原范畴组成的完全同态象类的具体形式和范畴为J－半单范畴的充要条件。     

On the Recollements of Functor Categories

This paper is devoted to the study of recollements of functor categories in different levels. In the first part of the paper, we start with a small category \(\mathcal {S}\) and a maximal object s of \(\mathcal {S}\) and construct a recollement of \(\text {Mod-}\mathcal {S}\) in terms of \(\text {Mod-End}_{\mathcal {S}}(s)\) and \(\text {Mod-}(\mathcal {S}\setminus \{s\})\) in four different levels. In case \(\mathcal {S}\) is a finite directed category, by iterating this argument, we get chains of recollements having some interesting applications. In the second part, we start with a recollement of rings and construct a recollement of their path rings, with respect to a finite quiver. Third part of the paper presents some applications, including recollements of triangular matrix rings, an example of a recollement in Gorenstein derived level and recollements of derived categories of N-complexes.     

Classification of Spherical Fusion Categories of Frobenius-Schur Exponent 2

In this paper,we propose a new approach towards the classification of spherical fusion categories by their Frobenius-Schur exponents.We classify spherical fusion categories of Frobenius-Schur exponent 2 up to monoidal equivalence.We also classify modular categories of Frobenius-Schur exponent 2 up to braided monoidal equivalence.It turns out that the Gauss sum is a complete invariant for modular categories of FrobeniusSchur exponent 2.This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.     

On Combinatorial Model Categories

Combinatorial model categories were introduced by J. H. Smith as model categories which are locally presentable and cofibrantly generated. He has not published his results yet but proofs of some of them were presented by T. Beke, D. Dugger or J. Lurie. We are contributing to this endeavour by some new results about homotopy equivalences, weak equivalences and cofibrations in combinatorial model categories. Supported by MSM 0021622409 and GAČR 201/06/0664.     

On Homogeneous Exact Categories

In the present paper we prove that a certain subcategory of the module category over some infinite-dimensional algebra R has almost split sequences and strongly homogeneous property; i.e., for each indecomposable module M in , there is an almost split sequence starting and also ending at M. It is also proved that except for a trivial case, is of wild representation type.     

关于几乎可解群

本文推广了关于局部有限群的Asar定理及p.Hall—Kulatilaka,Kargapolov定理.     

On Almost Unit-Regular Rings

An element a ∈ R is unit-regular provided that there exists an invertible u ∈ R such that a = aua. A ring R is called an almost unit-regular ring provided that for any a ∈ R, either a or 1 ? a is unit-regular. We characterize, in this article, the almost unit-regularity of Morita contexts with zero pairings. We also show that a ring R is unit-regular if and only if M ₂(R) is almost unit-regular. Various examples of such rings are constructed by means of formal triangular matrix rings.