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27
Double Lie algebroids and secondorder geometry
 I. Adv. Math
, 1992
"... We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We extend the construction of the tangent prolongation of an abstract Lie algebroid to show that the Lie algebroid structure of any LAgroupoid may be prolonged to the Lie algebr ..."
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Cited by 36 (5 self)
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We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We extend the construction of the tangent prolongation of an abstract Lie algebroid to show that the Lie algebroid structure of any LAgroupoid may be prolonged to the Lie algebroid of its groupoid structure. In the case of a double groupoid, this prolonged structure for either LAgroupoid is canonically isomorphic to the Lie algebroid structure associated with the other; this extends many canonical isomorphisms associated with iterated tangent and cotangent structures. We calculate several examples from Poisson geometry. We show that the cotangent of any double Lie groupoid is a symplectic double groupoid and that the side groupoids of a symplectic double groupoid are Poisson groupoids in duality; thus the duals of the LAgroupoids of any double groupoid are a pair of Poisson groupoids in duality.
Fibrations of groupoids
 J. Algebra
, 1970
"... theory, and change of base for groupoids and multiple ..."
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Cited by 29 (15 self)
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theory, and change of base for groupoids and multiple
Integration of Lie bialgebroids
, 1997
"... We prove that under certain mild assumptions a Lie bialgebroid integrates to a Poisson groupoid. This includes, in particular, a new proof of the existence of local symplectic groupoids for any Poisson manifold, a theorem of Karasev and of Weinstein. ..."
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Cited by 28 (5 self)
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We prove that under certain mild assumptions a Lie bialgebroid integrates to a Poisson groupoid. This includes, in particular, a new proof of the existence of local symplectic groupoids for any Poisson manifold, a theorem of Karasev and of Weinstein.
Crossed complexes and homotopy groupoids as noncommutative tools for higher dimensional localtoglobal problems
 In, ‘Proceedings of the Fields Institute Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras and Semiabelian Categories’ (September 2328, 2002), Fields Institute Communications 43
, 2004
"... ..."
Pseudo algebras and pseudo double categories
 J. Homotopy Relat. Struct
"... Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, an ..."
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Cited by 17 (2 self)
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Abstract. As an example of the categorical apparatus of pseudo algebras over 2theories, we show that pseudo algebras over the 2theory of categories can be viewed as pseudo double categories with folding or as appropriate 2functors into bicategories. Foldings are equivalent to connection pairs, and also to thin structures if the vertical and horizontal morphisms coincide. In a sense, the squares of a double category with folding are determined in a functorial way by the 2cells of the horizontal 2category. As a special case, strict 2algebras with one object and everything invertible are crossed modules under a group.
Covering groups of nonconnected topological groups, and the monodromy groupoid of a topological group
, 1993
"... All spaces are assumed to be locally path connected and semilocally 1connected. Let X be a connected topological group with identity e, and let p: ˜ X → X be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for a ..."
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Cited by 15 (11 self)
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All spaces are assumed to be locally path connected and semilocally 1connected. Let X be a connected topological group with identity e, and let p: ˜ X → X be the universal cover of the underlying space of X. It follows easily from classical properties of lifting maps to covering spaces that for any point ˜e in ˜ X with p˜e = e there is a unique structure of topological group on ˜ X such that ˜e is the
Drinfel’d doubles and Ehresmann doubles for Lie algebroids and Lie
, 1998
"... Abstract. We show that the Manin triple characterization of Lie bialgebras in terms of the Drinfel’d double may be extended to arbitrary Poisson manifolds and indeed Lie bialgebroids by using double cotangent bundles, rather than the direct sum structures (Courant algebroids) utilized for similar p ..."
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Cited by 13 (1 self)
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Abstract. We show that the Manin triple characterization of Lie bialgebras in terms of the Drinfel’d double may be extended to arbitrary Poisson manifolds and indeed Lie bialgebroids by using double cotangent bundles, rather than the direct sum structures (Courant algebroids) utilized for similar purnotion of double Lie algebroid (where \double " is now used in the Ehresmann sense) which unies many iterated constructions in dierential geometry. 1.
Geometrical mechanics on algebroids
 Int. J. Geom. Methods Mod. Phys
"... A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this app ..."
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Cited by 11 (2 self)
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A natural geometric framework is proposed, based on ideas of W. M. Tulczyjew, for constructions of dynamics on general algebroids. One obtains formalisms similar to the Lagrangian and the Hamiltonian ones. In contrast with recently studied concepts of Analytical Mechanics on Lie algebroids, this approach requires much less than the presence of a Lie algebroid structure on a vector bundle, but it still reproduces the main features of the Analytical Mechanics, like the EulerLagrangetype equations, the correspondence between the Lagrangian and Hamiltonian functions (Legendre transform) in the hyperregular cases, and a version of the Noether Theorem. Besides, the constructions seem to be more natural and simpler.
Homotopy Theory, and Change of Base for Groupoids and Multiple Groupoids
, 1996
"... This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids. ..."
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Cited by 7 (6 self)
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This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids.
Measurable Kac cohomology for Bicrossed Products
, 2003
"... We study the Kac cohomology for matched pairs of locally compact groups. This cohomology theory arises from the extension theory of locally compact quantum groups. We prove a topological version of the Kac exact sequence and provide methods to compute the cohomology. We give explicit calculations in ..."
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Cited by 6 (0 self)
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We study the Kac cohomology for matched pairs of locally compact groups. This cohomology theory arises from the extension theory of locally compact quantum groups. We prove a topological version of the Kac exact sequence and provide methods to compute the cohomology. We give explicit calculations in several examples using results of Moore and Wigner.