Groupoid Structure Research Articles

The structure of double groupoids

We give a general description of the structure of a discrete double groupoid (with an extra, quite natural, filling condition) in terms of groupoid factorizations and groupoid 2-cocycles with coefficients in abelian group bundles. Our description goes as follows: to any double groupoid, we associate an abelian group bundle and a second double groupoid, its frame. The frame satisfies that every box is determined by its edges, and thus is called a ‘slim’ double groupoid. In the first step, we prove that every double groupoid is obtained as an extension of its associated abelian group bundle by its frame. In the second, independent, step we prove that every slim double groupoid with a filling condition is completely determined by a factorization of a certain canonically defined ‘diagonal’ groupoid.

A class of Garside groupoid structures on the pure braid group

We construct a class of Garside groupoid structures on the pure braid groups, one for each function (called labelling) from the punctures to the integers greater than 1. The object set of the groupoid is the set of ball decompositions of the punctured disk; the labels are the perimeters of the regions. Our construction generalises Garside's original Garside structure, but not the one by Birman-Ko-Lee. As a consequence, we generalise the Tamari lattice ordering on the set of vertices of the associahedron.

GenOVa: a computer program to generate orientational variants

A computer program called GenOVa, written in Python, calculates the orientational variants, the operators (special types of misorientations between variants) and the composition table associated with a groupoid structure. The variants can be represented by three-dimensional shapes or by pole figures.

On the geometry of prequantization spaces

Given a Poisson (or more generally Dirac) manifold P , there are two approaches to its geometric quantization: one involves a circle bundle Q over P endowed with a Jacobi (or Jacobi–Dirac) structure; the other one involves a circle bundle with a (pre)contact groupoid structure over the (pre)symplectic groupoid of P . We study the relation between these two prequantization spaces. We show that the circle bundle over the (pre)symplectic groupoid of P is obtained from the Lie groupoid of Q via an S 1 reduction that preserves both the Lie groupoid and the geometric structures.

Multiple twinning in cubic crystals: geometric/algebraic study and its application for the identification of the Σ3ngrain boundaries

Multiple twinning in cubic crystals is represented geometrically by a three-dimensional fractal and algebraically by a groupoid. In this groupoid, the variant crystals are the objects, the misorientations between the variants are the operations, and the Sigma3(n) operators are the different types of operations (expressed by sets of equivalent operations). A general formula gives the number of variants and the number of Sigma3(n) operators for any twinning order. Different substructures of this groupoid (free group, semigroup) can be equivalently introduced to encode the operations with strings. For any coding substructure, the operators are expressed by sets of equivalent strings. The composition of two operators is determined without any matrix calculation by string concatenations. It is multivalued due to the groupoid structure. The composition table of the operators is used to identify the Sigma3(n) grain boundaries and to reconstruct the twin related domains in the electron back-scattered diffraction maps.

Reconstruction of parent grains from EBSD data

The limits of the parent grain reconstruction by the neighbor-to-neighbor method are shown. A new method of parent grain reconstruction from EBSD data is proposed for any phase transformation. It is based on the theoretical groupoid structure formed by the variants and their operators. It tolerates materials with high levels of intragranular deformation. It is quick and robust because it does not imply solving any equation, and mainly consists in the comparison of numbers. It has satisfactorily been applied to the reconstruction of austenitic grains in martensitic steels.

On the de Rham cohomology of differential and algebraic stacks

We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack determines the flat groupoid up to étale equivalence. We show how a cofoliation on a stack gives rise to a refinement of the Hodge to De Rham spectral sequence, where the E 1 -term consists entirely of vector bundle valued cohomology groups. Our theory works for differentiable, holomorphic and algebraic stacks.

Hyperelliptic Addition Law

Given a family of genus g algebraic curves, with the equation f(x, y, Λ)=0, we consider two fiber-bundles U and X over the space of parameters Λ. A fiber of U is the Jacobi variety of the curve. U is equipped with the natural groupoid structure that induces the canonical addition on a fiber. A fiber of X is the g-th symmetric power of the curve. We describe the algebraic groupoid structure on X using the Weierstrass gap theorem to define the ‘addition law’ on its fiber. The addition theorems that are the subject of the present study are represented by the formulas, mostly explicit, determining the isomorphism of groupoids U → X. At g=1 this gives the classic addition formulas for the elliptic Weierstrass ℘ and ℘′ functions. To illustrate the efficiency of our approach the hyperelliptic curves of the form are considered. We construct the explicit form of the addition law for hyperelliptic Abelian vector functions ℘ and ℘′ (the functions ℘ and ℘′ form a basis in the field of hyperelliptic Abelian functions, i.e., any function from the field can be expressed as a rational function of ℘ and ℘′). Addition formulas for the higher genera zetafunctions are discussed. The genus 2 result is written in a Hirota-like trilinear form for the sigma-function. We propose a conjecture to describe the general formula in these terms.

Interior symmetry and local bifurcation in coupled cell networks

A coupled cell system is a network of dynamical systems, or ‘cells’, coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells and edges that preserves all internal dynamics and all couplings. It is well known that symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. Recently, the introduction of a less stringent form of symmetry, the ‘symmetry groupoid’, has shown that global group-theoretic symmetry is not the only mechanism that can create such states in a coupled cell system. The symmetry groupoid consists of structure-preserving bijections between certain subsets of the cell network, the input sets. Here, we introduce a concept intermediate between the groupoid symmetries and the global group symmetries of a network: ‘interior symmetry’. This concept is closely related to the groupoid structure, but imposes stronger constraints of a group-theoretic nature. We develop the local bifurcation theory of coupled cell systems possessing interior symmetries, by analogy with symmetric bifurcation theory. The main results are analogues for ‘synchrony-breaking’ bifurcations of the Equivariant Branching Lemma for steady-state bifurcation, and the Equivariant Hopf Theorem for bifurcation to time-periodic states.

On Weak Regular *-semigroups

In this paper, a special kind of partial algebras called projective partial groupoids is defined. It is proved that the inverse image of all projections of a fundamental weak regular *-semigroup under the homomorphism induced by the maximum idempotent-separating congruence of a weak regular *-semigroup has a projective partial groupoid structure. Moreover, a weak regular *-product which connects a fundamental weak regular *-semigroup with corresponding projective partial groupoid is defined and characterized. It is finally proved that every weak regular *-product is in fact a weak regular *-semigroup and any weak regular *-semigroup is constructed in this way.

Dirac submanifolds and Poisson involutions

Dirac submanifolds are a natural generalization in the Poisson category of the symplectic submanifolds of a symplectic manifold. They correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable loci of Poisson involutions. In this paper, we make a general study of these submanifolds including both local and global aspects. In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable loci. In particular, we discuss the Poisson involutions on a special class of Poisson groups, and more generally Poisson groupoids, called symmetric Poisson groups, and symmetric Poisson groupoids. Many well-known examples, including the standard Poisson group structures on semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie groups, and Poisson groupoid structures arising from dynamical r-matrices of semi-simple Lie algebras are symmetric, so they admit a Poisson involution. For symmetric Poisson groups, the relation between the stable locus Poisson structure and Poisson symmetric spaces is discussed. As a consequence, we prove that the Dubrovin Poisson structure on the space of Stokes matrices U + (appearing in Dubrovin's theory of Frobenius manifolds) is a Poisson symmetric space.

LOOPS AND DISKS IN SURFACES AND HANDLEBODIES

We consider a groupoid structure involving graphs and imbedded circles (resp. disks) in a closed orientable surface F (resp. the handleboby H it bounds). We look at generalized connected sums and related operations "induced" by certain morphisms, and introduce a quandle structure involving isotopy classes of imbedded circles and Dehn twists in F. Some relations between these operations are examined, and we look at characterizations of imbedded disks in H arising in this algebraic context.

Every orientable 3–manifold is a BΓ

We show that every orientable 3-manifold is a classifying space B\Gamma where \Gamma is a groupoid of germs of homeomorphisms of R. This follows by showing that every orientable 3-manifold M admits a codimension one foliation F such that the holonomy cover of every leaf is contractible. The F we construct can be taken to be C^1 but not C^2. The existence of such an F answers positively a question posed by Tsuboi [Classifying spaces for groupoid structures, notes from minicourse at PUC, Rio de Janeiro (2001)], but leaves open the question of whether M = B\Gamma for some C^\infty groupoid \Gamma.

Finite simple zeropotent paramedial groupoids

The study of paramedial groupoids (with Emphasis on the structure of simple paramedial groupoids) was initiated in [1] and continued in [2], [3] and [5]. The aim of the present paper is to give a full description of finite simple zeropotent paramedial groupoids (i.e., of finite simple paramedial groupoids of Type (II)—see [2]). A reader is referred to [1], [2], [3] and [7] for notation and various prerequisites.

Symplectic areas, quantization, and dynamics in electromagnetic fields

A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: A choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the quantum evolution problem for a charged particle in an electromagnetic field is represented in an exact continual form and in the semiclassical approximation via the area of dynamical membranes.

Extendibility, monodromy, and local triviality for topological groupoids

A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all maps of groupoid structure are continuous. The notion of monodromy groupoid of a topological groupoid generalizes those of fundamental groupoid and universal cover. It was earlier proved that the monodromy groupoid of a locally sectionable topological groupoid has the structure of a topological groupoid satisfying some properties. In this paper a similar problem is studied for compatible locally trivial topological groupoids.

A Unified Approach to Poisson Reduction

Given any Poisson action G×P→P of a Poisson–Lie group G we construct an object Ω=T*G*T*P which has both a Lie groupoid structure and a Lie algebroid structure and which is a half-integrated form of the matched pair of Lie algebroids which J.-H. Lu associated to a Poisson action in her development of Drinfeld's classification of Poisson homogeneous spaces. We use Ω to give a general reduction procedure for Poisson group actions, which applies in cases where a moment map in the usual sense does not exist. The same method may be applied to actions of symplectic groupoids and, most generally, to actions of Poisson groupoids.

ON SYMPLECTIC DOUBLE GROUPOIDS AND THE DUALITY OF POISSON GROUPOIDS

We prove that the cotangent of a double Lie groupoid S has itself a double groupoid structure with sides the duals of associated Lie algebroids, and double base the dual of the Lie algebroid of the core of S. Using this, we prove a result outlined by Weinstein in 1988, that the side groupoids of a general symplectic double groupoid are Poisson groupoids in duality. Further, we prove that any double Lie groupoid gives rise to a pair of Poisson groupoids (and thus of Lie bialgebroids) in duality. To handle the structures involved effectively we extend to this context the dualities and canonical isomorphisms for tangent and cotangent structures of the author and Ping Xu.

Groupoid structure of Toeplitz algebra over L-shaped domain

The structure of the groupoidG associated with the ToeplitzC*-algebraC*(Ω) of the L-shaped domain is discussed. The detailed characterization ofM∞ by the classification of the closed subgroup of the Euclidean space is presented.

The Topological Structure of Polish Groups and Groupoids of Measure Space Transformations

It is proved that the groupoid of nonsingular partial isomorphisms of a Lebesgue space (X,μ) is weakly contractible in a “strong” sense: we present a contraction path which preserves invariant the subgroupoid of μ -preserving partial isomorphisms as well as the group of nonsingular transformations of X . Moreover, let \mathcal R be an ergodic measured discrete equivalence relation on X . The full group [\mathcal R] endowed with the uniform topology is shown to be contractible. For an approximately finite \mathcal R of type \mathit{II} or \mathit{III}_λ , 0≤λ<1 , the normalizer N[\mathcal R] of \mathcal R furnished with the natural Polish topology is established to be homotopically equivalent to the centralizer of the associated Poincaré flow. These are the measure theoretical analogues of the resent results of S. Popa and M. Takesaki on the topological structure of the unitary and the automorphism group of a factor.