The Muhly–Renault–Williams Theorem for Lie Groupoids and its Classical Counterpart

A theorem of Muhly–Renault–Williams states that if two locally compact groupoids with Haar system are Morita equivalent, then their associated convolution C*-algebras are strongly Morita equivalent. We give a new proof of this theorem for Lie groupoids. Subsequently, we prove a counterpart of this theorem in Poisson geometry: If two Morita equivalent Lie groupoids are s-connected and s-simply connected, then their associated Poisson manifolds (viz. the dual bundles to their Lie algebroids) are Morita equivalent in the sense of P. Xu.     

Locally Vacant Double Lie Groupoids and the Integration of Matched Pairs of Lie Algebroids

We prove a general integrability result for matched pairs of Lie algebroids. (Matched pairs of Lie algebras are also known as double Lie algebras or twilled extensions of Lie algebras.) The method used is an extension of a method introduced by Lu and Weinstein in the case of Poisson Lie groups, and yields double groupoids which satisfy an étale form of the vacancy condition.     

Prime étale groupoid algebras with applications to inverse semigroup and Leavitt path algebras

In this paper we give some sufficient and some necessary conditions for an étale groupoid algebra to be a prime ring. As an application we recover the known primeness results for inverse semigroup algebras and Leavitt path algebras. It turns out that primeness of the algebra is connected with the dynamical property of topological transitivity of the groupoid. We obtain analogous results for semiprimeness.     

Matrices, graphs and equivalence relations

A suitable equivalence relation is introduced on the set of square matrices with entries of any kind. This allows us to associate to every equivalence class an infinite family of graphs and determine their topological properties. When a given square matrix is the multiplication table of a finite groupoid, some connections between algebraic properties of the groupoid and topological properties of these graphs are proved. Received: May 4, 1999 Published online: December 19, 2001     

Combinatorial Groupoids,Cubical Complexes,and the Lovász Conjecture

This paper lays the foundation for a theory of combinatorial groupoids that allows us to use concepts like “holonomy”, “parallel transport”, “bundles”, “combinatorial curvature”, etc. in the context of simplicial (polyhedral) complexes, posets, graphs, polytopes and other combinatorial objects. We introduce a new, holonomy-type invariant for cubical complexes, leading to a combinatorial “Theorema Egregium” for cubical complexes that are non-embeddable into cubical lattices. Parallel transport of Hom-complexes and maps is used as a tool to extend Babson–Kozlov–Lovász graph coloring results to more general statements about nondegenerate maps (colorings) of simplicial complexes and graphs. The author was supported by grants 144014 and 144026 of the Serbian Ministry of Science and Technology.     

Special Symplectic Connections and Poisson Geometry

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that any special symplectic connection can be constructed using symplectic realizations of quadratic deformations of a certain linear Poisson structure. Moreover, we show that these Poisson structures cannot be symplectically integrated by a Hausdorff groupoid. As a consequence, we obtain a canonical principal line bundle over any special symplectic manifold or orbifold, and we deduce numerous global consequences.     

K-Theory of the Indicial Algebra of a Manifold with Corners

We compute the K-theory, groups of the C ^*-algebra of the groupoid of a manifold with corners, in which the analytic index takes its values.     

单纯广群的准层和同伦纤维

显示了在设置C上的单纯广群的准层的范畴是个封闭模型范畴.证明了在一个单纯广群的准层G上的单纯函子X是局部弱等价于同伦纤维.