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Quantization as a functor
 in ”Quantization, Poisson Brackets and beyond”, Contemp. Math.,315, AMS
, 2002
"... “First quantization is a mystery, but second quantization is a functor” ..."
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“First quantization is a mystery, but second quantization is a functor”
Stacky Lie Groups
, 2008
"... Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary ..."
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Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as group objects in this weak 2category. Introducing a graphic notation, it is shown that for every stacky Lie monoid there is a natural morphism, called the preinverse, which is a Morita equivalence if and only if the monoid is a stacky Lie group. As an example, we describe explicitly the stacky Lie group structure of the irrational Kronecker foliation of the torus.
Crystallographic Topology 2: Overview And Work In Progress
, 1999
"... This overview describes an application of contemporary geometric topology and stochastic process concepts to structural crystallography. In this application, crystallographic groups become orbifolds, crystal structures become Morse functions on orbifolds, and vibrating atoms in a crystal become vect ..."
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This overview describes an application of contemporary geometric topology and stochastic process concepts to structural crystallography. In this application, crystallographic groups become orbifolds, crystal structures become Morse functions on orbifolds, and vibrating atoms in a crystal become vector valued Gaussian measures with the RadonNikodym property. Intended crystallographic benefits include new methods for visualization of space groups and crystal structures, analysis of the thermal motion patterns seen in ORTEP drawings, and a classification scheme for crystal structures based on their Heegaard splitting properties. 1 Introduction Geometric topology and structural crystallography concepts are combined to define a research area we call Structural Crystallographic Topology, or just Crystallographic Topology. The first paper in the series[30] describes basic crystallography concepts (crystallographic groups, lattice complexes, and crystal structures) and their replacement topo...
Simplicial Cohomology of Smooth Orbifolds in GAP
"... Abstract. This short research announcement briefly describes the simplicial method underlying the GAP package SCO for computing the socalled orbifold cohomology of topological resp. smooth orbifolds. SCO can be used to compute the lower dimensional group cohomology of some infinite groups. Instead ..."
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Abstract. This short research announcement briefly describes the simplicial method underlying the GAP package SCO for computing the socalled orbifold cohomology of topological resp. smooth orbifolds. SCO can be used to compute the lower dimensional group cohomology of some infinite groups. Instead of giving a complete formal definition of an orbifold, we start with a simple construction, general enough to give rise to any orbifold M. Let X be a (smooth) manifold and G a Lie group acting (smoothly and) properly on X, i.e., the action graph α: G×X → X ×X: (g, x) 7 → (x, gx) is a proper map. In particular, G acts with compact stabilizers Gx = α −1({(x, x)}). Further assume that G is either – discrete (acting discontinuously), or – compact acting almost freely (i.e. with discrete stabilizers) on X. In both cases G acts with finite stabilizers. “Enriching ” M: = X/G with these
Functoriality and Morita equivalence of operator algebras and Poisson manifolds associated to groupoids
, 2008
"... ..."
DRINFELD DOUBLE FOR ORBIFOLDS
, 2005
"... Abstract. We prove that the Drinfeld double of the category of sheaves on ..."
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Abstract. We prove that the Drinfeld double of the category of sheaves on
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
, 2003
"... Abstract. It is wellknown that an effective orbifold M (one for which the local stabilizer groups act effectively) can be presented as a quotient of a smooth manifold P by a locally free action of a compact lie group K. We use the language of groupoids to provide a partial answer to the question of ..."
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Abstract. It is wellknown that an effective orbifold M (one for which the local stabilizer groups act effectively) can be presented as a quotient of a smooth manifold P by a locally free action of a compact lie group K. We use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented. We also note some connections to stacks and gerbes. 1.