Coherence for Product Monoids and their Actions

Let and be two monoids (algebras) in a monoidal category . Further let be a distributive law in the sense of [J. Beck, Lect. Notes Math., 80:119–140, 1969]; naturally yields a monoid . Consider a word in the symbols , , and . The first coherence theorem proved in this paper asserts that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , and . Assume now that an object is endowed with both an -object structure , and an -object structure . Further assume that these two structures are compatible, in the sense that they naturally yield an -object . Let be a word in , , , and , which contains a single instance of , in the rightmost position. The second coherence theorem states that all morphisms coincide in , provided they arise as composites of morphisms which are -products of ’s ‘canonical’ structure morphisms, and of , , , , , , , , , and .     

Relations between different types of spectra and spectral characterizations

In the study of the asymptotic behaviour of solutions of differential-difference equations the -spectrum has been useful, where and implies Fourier transform , with given , φ∈L ^∞(ℝ,X), X a Banach space, (half)line. Here we study and related concepts, give relations between them, especially weak Laplace half-line spectrum of φ, and thus ⊂ classical Beurling spectrum = Carleman spectrum =  ; also  = Beurling spectrum of “φ modulo ” (Chill-Fasangova). If satisfies a Loomis type condition (L _U ), then countable and uniformly continuous ∈U are shown to imply ; here (L _U ) usually means , indefinite integral Pf of f in U imply Pf in (the Bohl-Bohr theorem for = almost periodic functions, U=bounded functions). This spectral characterization and other results are extended to unbounded functions via mean classes , ℳ ^m U ((2.1) below) and even to distributions, generalizing various recent results for uniformly continuous bounded φ. Furthermore for solutions of convolution systems S*φ=b with in some we show . With these above results, one gets generalizations of earlier results on the asymptotic behaviour of solutions of neutral integro-differential-difference systems. Also many examples and special cases are discussed.     

Generalized Continuous Posets and a New Cartesian Closed Category

With every subset selection for posets, there is associated a certain ideal completion . As shown by Erné, such completions help to extend classical results on domains and similar structures in the absence of the required joins. Some results about –predistributive or –precontinuous posets and –continuous functions are summarized and supplemented. In particular, several central results on function spaces in domain theory are extended to the setting of productive closed subset selections. The category FSBP, in which objects are finitely separated and upper bounded posets and arrows are continuous functions between them, is shown to be cartesian closed. This research is supported by the National Natural Science Foundation of China, 10471035.     

\boldsymbol{\mathcal{Q}}-*-Categories

We consider the theory of categories enriched in an involutive quantaloid : the -*-categories. After giving an introduction to involutive quantaloids and nuclei, we use matrices with entries in to define -*-categories. Then we examine the relations between two kinds of morphisms between them, the functors and the *-maps, to provide a basis to study completeness properties. These results are used to provide a definition of pseudo-presheaves, presheaves and sheaves on involutive quantaloids in order to get a generalization of presheaves and sheaves on sites. Finally a characterization of these sheaves in terms of covers and compatible families is presented.      

Admissible Galois Structures and Coverings in Regular Mal’cev Categories

Given a regular Gumm category such that any regular epimorphism is effective for descent, we prove that any Birkhoff subcategory in gives rise to an admissible Galois structure. This result allows one to consider some new applications of the categorical Galois theory in the context of topological algebras. Given a regular Mal’cev category , we first characterize the coverings of the Galois structure induced by the subcategory of the abelian objects in . Then we consider as a subcategory of the category of the equivalence relations in , and we characterize the coverings of the corresponding Galois structure . By composing the Galois structures and we obtain the Galois structure induced by as a subcategory of . We give the characterization of the -coverings in terms of the coverings of and .     

A Relation Between Closure Operators on a Small Category and Its Category of Presheaves

In this paper we show that each factorization structure on a small category , satisfying certain conditions, yields a presheaf on and a morphism of presheaves . We then give connections, and set up one to one correspondences, between subclasses of the following classes: (a) closure operators on (b) subobjects of (c) morphisms from to (d) weak Lawvere–Tierney topologies (e) weak Grothendieck topologies (f) closure operators on .     

Sober Approach Spaces are Firmly Reflective for the Class of Epimorphic Embeddings

In [2] the subconstruct of sober approach spaces was introduced and it was shown to be a reflective subconstruct of the category of approach spaces. The main result of this paper states that moreover is firmly -reflective in for the class of epimorphic embeddings. ‘Firm -reflective’ is a notion introduced in [3] by G.C.L. Brümmer and E. Giuli and is inspired by the exemplary behaviour of the usual completion in the category of Hausdorff uniform spaces with uniformly continuous maps. It means that is -reflective in and that the reflector is such that belongs to if and only if is an isomorphism. Firm -reflectiveness implies uniqueness of completion in the sense that whenever is a map with and sober, the associated is an isomorphism. Our result generalizes the fact that in the category the subconstruct of sober topological spaces is firmly reflective for the class of b-dense embeddings in . Also firmness in some other subconstructs of will be easily obtained.A. Gerlo and C. Van Olmen are research assistants at the Fund of Scientific Research Vlaanderen (FWO). E. Vandersmissen is a research assistant supported by the FWO-grant G.0244.05.     

Nested set complexes of Dowling lattices and complexes of Dowling trees

Given a finite group G and a natural number n, we study the structure of the complex of nested sets of the associated Dowling lattice (Proc. Internat. Sympos., 1971, pp. 101–115) and of its subposet of the G-symmetric partitions which was recently introduced by Hultman (, 2006), together with the complex of G-symmetric phylogenetic trees . Hultman shows that the complexes and are homotopy equivalent and Cohen–Macaulay, and determines the rank of their top homology. An application of the theory of building sets and nested set complexes by Feichtner and Kozlov (Selecta Math. (N.S.) 10, 37–60, 2004) shows that in fact is subdivided by the order complex of . We introduce the complex of Dowling trees and prove that it is subdivided by the order complex of . Application of a theorem of Feichtner and Sturmfels (Port. Math. (N.S.) 62, 437–468, 2005) shows that, as a simplicial complex, is in fact isomorphic to the Bergman complex of the associated Dowling geometry. Topologically, we prove that is obtained from by successive coning over certain subcomplexes. It is well known that is shellable, and of the same dimension as . We explicitly and independently calculate how many homology spheres are added in passing from to . Comparison with work of Gottlieb and Wachs (Adv. Appl. Math. 24(4), 301–336, 2000) shows that is intimely related to the representation theory of the top homology of . Research partially supported by the Swiss National Science Foundation, project PP002-106403/1.     

The Fusion Algebra of Bimodule Categories

We establish an algebra-isomorphism between the complexified Grothendieck ring of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of . As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category.      

Categorical Structures Enriched in a Quantaloid: Orders and Ideals over a Base Quantaloid

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid , which we call ‘ -order’. This requires a theory of semicategories enriched in the quantaloid , that admit a suitable Cauchy completion. There is a quantaloid of -orders and ideal relations, and a locally ordered category of -orders and monotone maps; actually, . In particular is , with Ω a locale, the category of ordered objects in the topos of sheaves on Ω. In general -orders can equivalently be described as Cauchy complete categories enriched in the split-idempotent completion of . Applied to a locale Ω this generalizes and unifies previous treatments of (ordered) sheaves on Ω in terms of Ω-enriched structures.Mathematics Subject Classifications (2000) 06F07, 18B35, 18D05, 18D20.     

Hilbert Space Representations of Cross Product Algebras II

In this paper, we study and classify Hilbert space representations of cross product -algebras of the quantized enveloping algebra with the coordinate algebras of the quantum motion group and of the complex plane, and of the quantized enveloping algebra with the coordinate algebras of the quantum group and of the quantum disc. Invariant positive functionals and the corresponding Heisenberg representations are explicitly described.Presented by S.L. Woronowicz.     

Dimension of the Torelli group for Out(F<Subscript>n</Subscript>)

Let be the kernel of the natural map Out(F_n)→GL_n(ℤ). We use combinatorial Morse theory to prove that has an Eilenberg–MacLane space which is (2n-4)-dimensional and that is not finitely generated (n≥3). In particular, this shows that the cohomological dimension of is equal to 2n-4 and recovers the result of Krstić–McCool that is not finitely presented. We also give a new proof of the fact, due to Magnus, that is finitely generated.     

Epi-topology and Epi-convergence for Archimedean Lattice-ordered Groups with Unit

is the category of archimedean -groups with distinguished weak order unit, with -group homomorphisms which preserve unit. This category includes all rings of continuous functions and all rings of measurable functions modulo null functions, with ring homomorphisms. The authors, and others, have studied previously the epimorphisms (right-cancellable morphisms) in . There is a rich theory. In this paper, we describe a topological approach to the analysis of these epimorphisms. On each – object, we define a topology and a convergence . These have the same closure operator, and this closure “captures epics” in the sense: a divisible subobject of is dense iff is epically embedded. The topology is , but only sometimes Hausdorff or an -group topology. The convergence is a Hausdorff -group convergence, but only sometimes topological. The associations of to , and to , are functorial. Dedicated to Bernhard Banaschewski for his 80th birthday.     

Primitive Ideals and Automorphisms of Quantum Matrices

Let be a field and q be a nonzero element of that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra of quantum matrices. Next, we describe all height one primes of ; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of whenever has only finitely many height one primes. Finally, we compute the automorphism group of in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of . Then, by using the preferred basis of and PBW bases, we prove that the automorphism group of is isomorphic to the torus when m ≠ n and (m,n) ≠ (1, 3),(3, 1). This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.     

Epicompletion in Frames with Skeletal Maps,I: Compact Regular Frames

A frame homomorphism h : A ⟶ B is skeletal if x ^⊥⊥ = 1 in A implies that h(x)^⊥⊥ = 1 in B. It is shown that, in , the category of compact regular frames with skeletal maps, the subcategory , consisting of the frames in which every polar is complemented, coincides with the epicomplete objects in . Further, is the least epireflective subcategory, and, indeed, the target of the monoreflection which assigns to a compact regular frame A, the ideal frame ε A of , the boolean algebra of polars of A.      

Slices of Essentially Algebraic Categories

This paper is a contribution to the theory of functor slices of J. Sichler and V. Trnková. For every ordinal α we introduce a basket , prove that every essentially algebraic category of height α is a slice of , characterize small slices of and give a common generalization of known results about slices of the algebraic basket .      

Representation theory of $\mathcal{W}$-algebras

We study the representation theory of the -algebra associated with a simple Lie algebra at level k. We show that the “-” reduction functor is exact and sends an irreducible module to zero or an irreducible module at any level k∈ℂ. Moreover, we show that the character of each irreducible highest weight representation of is completely determined by that of the corresponding irreducible highest weight representation of affine Lie algebra of . As a consequence we complete (for the “-” reduction) the proof of the conjecture of E. Frenkel, V. Kac and M. Wakimoto on the existence and the construction of the modular invariant representations of -algebras. Mathematics Subject Classification (1991)  17B68, 81R10     

On cyclotomic schemes over finite near-fields

We introduce a concept of cyclotomic association scheme over a finite near-field . It is proved that any isomorphism of two such nontrivial schemes is induced by a suitable element of the group AGL(V), where V is the linear space associated with . A sufficient condition on a cyclotomic scheme that guarantee the inclusion where is a finite field with elements, is given. I. Ponomarenko partially supported by RFFI, grants 03-01-00349, NSH-2251.2003.1.