The Categories of Yetter—Drinfel’d Modules, Doi—Hopf Modules and Two-sided Two-cosided Hopf Modules

We show that the two-sided two-cosided Hopf modules are in some case generalized Hopf modules in the sense of Doi. Then the equivalence between two-sided two-cosided Hopf modules and Yetter—Drinfeld modules, proved in [8], becomes an equivalence between categories of Doi—Hopf modules. This equivalence induces equivalences between the underlying categories of (co)modules. We study the relation between this equivalence and the one given by the induced functor.     

Tilting modules and ∗-modules

The relation between ?-modules-studied in [MO], [D], [C], [DH], [CM], [Z] and [T]-and Tiltng modules over an arbitrary ring is analyzed. In particular we prove that Tilting modules are exactly the faithful and finendo ?-modules. This answers a question of Trlifaj [T, Problem 1.5], showing that for any ring R the class of ?-modules generating the injectives and that one of Tiltings coincides. As a first application, we give an easy proof of the fact that every faithful ?-module over a finite-dimensional K-algebra is a classical Tilting module (see [DH, Theorem 1]). As a second application, we characterise the Tiltings as those modules which induce an equivalence between two categories with suitable dual properties.     

Weyl equivalence for rank 2 Nichols algebras of diagonal type

Yetter-Drinfel'd modules of diagonal type admit an equivalence relation which preserves dimension and Gel'fand-Kirillov dimension of the corresponding Nichols algebras. This relation is determined explicity for all rank 2 Yetter-Drinfel'd modules where the Gel'fand-Kirillov dimension is known to be finite. Supported by the European Community under a Marie Curie Intra-European Fellowship.     

On families of invariant lines of a Brouwer homeomorphism

ABSTRACT

We present properties of equivalence classes of the codivergency relation defined for a Brouwer homeomorphism for which there exists a family of invariant pairwise disjoint lines covering the plane. In particular, using the codivergency relation we describe the sets of regular and irregular points for such Brouwer homeomorphisms. Moreover, we show that under this assumption the interior of each equivalence class of this relation is invariant and simply connected.     

Classifying foliations of 3-manifolds via branched surfaces

We use branched surfaces to define an equivalence relation on C¹ codimension one foliations of any closed orientable 3-manifold that are transverse to some fixed nonsingular flow. There is a discrete metric on the set of equivalence classes with the property that foliations that are sufficiently close (up to equivalence) share important topological properties.     

Absolute Equivalence and Dirac Operators of Commuting Tuples of Operators

In this paper we define an equivalence relation of operators on Hilbert spaces which we call absolute equivalence. Two operators are called absolutely equivalent if both the absolute value of the operators and their adjoints are unitarily equivalent. We then use the properties of this equivalence relation to study the Koszul complex of a commuting tuple of operators through the Dirac operator of the tuple.     

Graphs realised by r.e. equivalence relations

We investigate dependence of recursively enumerable graphs on the equality relation given by a specific r.e. equivalence relation on ω. In particular we compare r.e. equivalence relations in terms of graphs they permit to represent. This defines partially ordered sets that depend on classes of graphs under consideration. We investigate some algebraic properties of these partially ordered sets. For instance, we show that some of these partial ordered sets possess atoms, minimal and maximal elements. We also fully describe the isomorphism types of some of these partial orders.     

Symmetries in synaptic algebras

A synaptic algebra is a generalization of the Jordan algebra of self-adjoint elements of a von Neumann algebra. We study symmetries in synaptic algebras, i.e., elements whose square is the unit element, and we investigate the equivalence relation on the projection lattice of the algebra induced by finite sequences of symmetries. In case the projection lattice is complete, or even centrally orthocomplete, this equivalence relation is shown to possess many of the properties of a dimension equivalence relation on an orthomodular lattice.     

Indecomposability of treed equivalence relations

We define rigorously a “treed” equivalence relation, which, intuitively, is an equivalence relation together with a measurably varying tree structure on each equivalence class. We show, in the nonamenable, ergodic, measure-preserving case, that a treed equivalence relation cannot be stably isomorphic to a direct product of two ergodic equivalence relations.     

Plactification

We study a map called plactification from reduced words to words. This map takes Coxeter-Knuth equivalence to Knuth equivalence, and has applications to the enumeration of reduced words, Schubert polynomials and certain Specht modules.     

Equivalence classes of functions between finite groups

Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals and splitting relative difference sets (RDSs) and introduce an intermediate relation, canonical equivalence, to aid in distinguishing the classes. We identify very precisely the conditions under which a graph equivalence determines a bundle equivalence, using transversals and extensions. We derive a new and easily computed algebraic measure of nonlinearity for a function f, calculated from the image of its coboundary ∂f. This measure is preserved by bundle equivalence but not by the coarser equivalences. It takes its minimum value if f is a homomorphism, and takes its maximum value if the graph of f contains a splitting RDS.     

Hopf modules and the double of a quasi-Hopf algebra

We give a different proof for a structure theorem of Hausser and Nill on Hopf modules over quasi-Hopf algebras. We extend the structure theorem to a classification of two-sided two-cosided Hopf modules by Yetter-Drinfeld modules, which can be defined in two rather different manners for the quasi-Hopf case. The category equivalence between Hopf modules and Yetter-Drinfeld modules leads to a new construction of the Drinfeld double of a quasi-Hopf algebra, as proposed by Majid and constructed by Hausser and Nill.

     

MODULES FOR WHICH EVERY SUBMODULE HAS A UNIQUE COCLOSURE

ABSTRACT

P. F. Smith studied modules in which every submodule has a unique closure and called them UC modules. In this paper we consider modules with the dual property viz., those in which every submodule has a unique coclosure and call such modules UCC modules. Unlike closures, a coclosure of a submodule of a module may not always exist and even if it exists, it may not be unique. We investigate the conditions under which a module is a UCC module. We prove that UCC modules are closed under factor modules and coclosed submodules. We also investigate their properties and their relation to non-cosingular modules, copolyform modules, and codimension modules. We end this paper with the dual of Smith's result on dimension modules.     

Trivial module for ortho-symplectic Lie superalgebras and Littlewood’s formula

We show that the denominator identity for ortho-symplectic Lie superalgebras osp(k|2n) is equivalent to the Littlewood’s formula. Such an equivalence also implies the relation between the trivial module and generalized Verma modules for osp(k|2n). Furthermore, we discuss the harmonic representative elements of the Kostant’s u-cohomology with trivial coefficients.     

Local equivalence relations

In this paper, we investigate the concept of local equivalence relation, a notion suggested by Grothendieck. A local equivalence relation on a topological space X is a global section of the sheaf of germs of equivalence relations on X. We investigate the extent to which a local equivalence relation can be described by a global one and analogously when can a global equivalence relation be recovered from its associated local one. We also look at the notion of a fiber map, which sheds further light on these concepts. A motivating example is that of a foliation on a manifold.     

On functors between categories of modules over trusses

Categorical aspects of the theory of modules over trusses are studied. Tensor product of modules over trusses is defined and its existence established. In particular, it is shown that bimodules over trusses form a monoidal category. Truss versions of the Eilenberg-Watts theorem and Morita equivalence are formulated. Projective and small-projective modules over trusses are defined and their properties studied.     

On certain vertex algebras and their modules associated with vertex algebroids

We study the family of vertex algebras associated with vertex algebroids, constructed by Gorbounov, Malikov, and Schechtman. As the main result, we classify all the (graded) simple modules for such vertex algebras and we show that the equivalence classes of graded simple modules one-to-one correspond to the equivalence classes of simple modules for the Lie algebroids associated with the vertex algebroids. To achieve our goal, we construct and exploit a Lie algebra from a given vertex algebroid.     

On properties of the set of invariant lines of a Brouwer homeomorphism

Abstract

We present properties of sets of invariant lines for Brouwer homeomorphisms which are not necessarily embeddable in a flow. Using such lines we describe the structure of equivalence classes of the codivergency relation. We also obtain a result concerning the set of regular points.     

Representations of étale Lie groupoids and modules over Hopf algebroids

The classical Serre-Swan’s theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita category of étale Lie groupoids and we show that the given correspondence represents a natural equivalence between them.     

Pure Anderson motives and abelian ��-sheaves

Pure t-motives were introduced by G. Anderson as higher dimensional generalizations of Drinfeld modules, and as the appropriate analogs of abelian varieties in the arithmetic of function fields. In order to construct moduli spaces for pure t-motives the second author has previously introduced the concept of abelian ??-sheaf. In this article we clarify the relation between pure t-motives and abelian ??-sheaves. We obtain an equivalence of the respective quasi-isogeny categories. Furthermore, we develop the elementary theory of both structures regarding morphisms, isogenies, Tate modules, and local shtukas. The later are the analogs of p-divisible groups.