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 th...

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann’s moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upo...

We associate to a Hausdorff space, X, a double groupoid, ρ 2 (X), the homotopy double groupoid of X. The construction is based on the geometric notion of thin square. Under the equivalence of categories between small 2-categories and double categories with connection given in [BM] the homotopy double groupoid corresponds to the homotopy 2-groupoid, G2(X), constructed in [HKK]. The cubical natur...

Suppose G is a second countable, locally compact, Hausdorff groupoid with a fixed left Haar system. Let G/G denote the orbit space of G and C∗(G) denote the groupoid C∗-algebra. Suppose that G is a principal groupoid. We show that C∗(G) is CCR if and only if G/G is a T1 topological space, and that C∗(G) is GCR if and only if G/G is a T0 topological space. We also show that C∗(G) is a Fell Algeb...

In this note a functorial approach to the integration problem of an LAgroupoid to a double Lie groupoid is discussed. To do that, we study the notions of fibered products in the categories of Lie groupoids and Lie algebroids, giving necessary and sufficient conditions for the existence of such. In particular, it turns out, that the fibered product of Lie algebroids along integrable morphisms is...

In [6–8] Murskii studied probability in algebra. He showed, for example, that the probability that a random groupoid is simple is 1 and that the probability that it has no idempotent element (one that satisfies a = a) is 1/e. To make these statements precise, define a groupoid table on the set {0, 1, . . . , n− 1} to be an n by n matrix with entries from {0, 1, . . . , n− 1}. Of course, each gr...

In this paper we study the existence of commuting regular elements, verifying the notion left (right) commuting regular elements and its properties in the groupoid G(n). Also we show that G(n) contains commuting regular subsemigroup and give a necessary and sufficient condition for the groupoid G(n) to be commuting regular.

By a 2-group we mean a groupoid equipped with a weakened group structure. It is called split when it is equivalent to the semidirect product of a discrete 2-group and a oneobject 2-group. By a permutation 2-group we mean the 2-group Sym(G) of self-equivalences of a groupoid G and natural isomorphisms between them, with the product given by composition of self-equivalences. These generalize the ...

For any body-time manifold [Formula: see text] there exists a groupoid, called the material encoding all properties of evolution. A smooth distribution, is constructed to deal with case in which groupoid not Lie groupoid. This new tool provides unified framework general non-uniform

X iv :0 80 3. 04 72 v1 [ m at h. R A ] 4 M ar 2 00 8 On decomposition of commutative Moufang groupoids B.V.Novikov It is well-known that the multiplicative groupoid of an alternative/Jordan algebra satisfies Moufang identities [1, 4]. Therefore it seems interesting to study the structure of such groupoids. In this note we apply to Moufang groupoids an approach which is widespread in Semigroup T...