Related papers: A Groupoid Structure on a Vector Space
The purpose of this paper is to study the transitive group-groupoids.
The purpose of this paper is to introduce the notion of loop groupoid associated to a groupoid. After studying the general properties of the loop groupoid, we show how this notion provides a very natural geometric interpretation for the…
A group, defined as set with associative multiplication and inverse, is a natural structure describing the symmetry of a space. The concept of group generalizes to group objects internal to other categories than sets. But there are yet more…
In this paper, we define a new structure analogous to group, called partial group. This structure concerns the partial stability by the composition inner law. We generalize the three isomorphism theorems for groups to partial groups.
We show that groupoid equivalence preserves a number of groupoid properties such as properness or the property of being topologically principal.
In this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor)…
The uniform structure on a differential space defined by a family of generators is considered.
We introduce a notion of topological property (T) for \'etale groupoids. This simultaneously generalizes Kazhdan's property (T) for groups and geometric property (T) for coarse spaces. One main goal is to use this property (T) to prove the…
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…
We give a modern account of the construction and structure of the space of generalized connections, an extension of the space of connections that plays a central role in loop quantum gravity.
In this article, we introduce a new object, a virtual quadratic space, and its group of isometries. They are presented as natural generalizations of quadratic spaces and orthogonal groups. It is then shown that by replacing quadratic spaces…
A spatial surface is a compact surface embedded in the $3$-sphere. We assume that a spatial surface is oriented and that each connected component of a spatial surface is neither a disk nor without a boundary. A diagram of a spatial surface…
In this paper we establish the principal bundle counterpart of the well-known and widely applied notion of vector bundle groupoid (VB-groupoid). In particular, we provide a general notion of principal bundle groupoid (PB-groupoid) as a…
We develop a generalised gauge theory in which the role of gauge group is played by a coalgebra and the role of principal bundle by an algebra. The theory provides a unifying point of view which includes quantum group gauge theory,…
In this paper, we study connections between the structure of a group and the structure of the group (under pointwise product) of its polynomial functions.
We give a brief overview of the properties of a higher dimensional generalization of matrix model which arises naturally in the context of a background independent approach to quantum gravity, the so called group field theory. We show that…
We describe the "generic" part of the character ring of general linear groups over a finite field in terms of quiver representations.
An objective of the theory of combinatorial groupoids is to introduce concepts like "holonomy", "parallel transport", "bundles", "combinatorial curvature" etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes,…
We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups.
A new generalization of Grassmannians, called {\nu}-grassmannians, and a canonical super vector bundle over this new space, say {\Gamma}, are introduced. Then, constructing a Gauss supermap of a super vector bundle, the universal property…