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In this paper we discuss generalized group, provides some interesting examples. Further we introduce a generalized module as a module like structure obtained from a generalized group and discuss some of its properties and we also describes…
We treat spectral problems by twisted groupoid methods. To Hausdorff locally compact groupoids endowed with a continuous $2$-cocycle one associates the reduced twisted groupoid $C^*$-algebra. Elements (or multipliers) of this algebra admit…
In this paper, we explore the theme of orbifold stratified spaces and establish a general criterion for them to be smooth orbifolds. This criterion utilizes the notion of linear stratification on the gluing bundles for the orbifold…
We show that the bicategory of (representable) orbifolds and good maps is equivalent to the bicategory of orbifold translation groupoids and generalized equivariant maps. We use this result to define an orbifold version of Bredon…
After attaching explicitly to the M\"obius strip an invertible module over the ring of real polynomial functions on the real circle, we expound as directly as possible the many faces and the main algebraic properties of invertible modules.…
This expository paper recounts the development and application of the concept of the diffeological groupoid, from its introduction in 1985 to its use in current research. We demonstrate how this single concept has served as a powerful and…
This paper introduces the notion of ``relative gerbes'' for smooth maps of manifolds, and discusses their differential geometry. The equivalence classes of relative gerbes are further classified by the relative integral cohomology in degree…
We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered $R$-categories, thus answering questions of what might be `commutative versions' of these theories.…
Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to…
For any unoriented loop on a compact connected oriented surface with one boundary component, the generalized Dehn twist along the loop is defined as an automorphism of the completed group ring of the fundamental group of the surface. If the…
The main purpose of this paper is to give a new definition for the notion of group-groupoid. Also, several basic properties of group-groupoids are established.
This paper classifies the derivations of twisted group algebras in terms of the generators and defining relations of the group. In particular, we generalize some know results over group algebras to the case of twisted group algebras. We…
We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear…
The article is devoted to the investigation of groups of diffeomorphisms and loops of manifolds over ultra-metric fields of zero and positive characteristics. Different types of topologies are considered on groups of loops and…
Motivated by orbifold string theory, we introduce orbifold cohomology group for any almost complex orbifold and orbifold Dolbeault cohomology for any complex orbifold. Then, we show that our new cohomology group satisfies Poincare duality…
As a branch of algebraic and differential topology of manifolds, the theory of Morse functions and their higher dimensional versions or fold maps and its application to algebraic and differential topology of manifolds is fundamental,…
Given the notion of suborbifold of the second author (based on ideas of Borzellino/Brunsden) and the classical correspondence (up to certain equivalences) between (effective) orbifolds via atlases and effective orbifold groupoids, we…
The theory of twistors on foliated manifolds is developed and the twistor space of the normal bundle is constructed. It is demonstrated that the classical constructions of the twistor theory lead to foliated objects and permit to formulate…
This paper describes a generalization of decomposition in orbifolds. In general terms, decomposition states that two-dimensional orbifolds and gauge theories whose gauge groups have trivially-acting subgroups decompose into disjoint unions…
We derive the basic correlation functions of twist fields coming from arbitrary twisted sectors in symmetric $Z_N$ orbifold conformal field theories, keeping all the admissible marginal perturbations, in particular those corresponding to…