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25
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 34 (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.
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 27 (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.
Fibrations of groupoids
 J. Algebra
, 1970
"... theory, and change of base for groupoids and multiple ..."
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Cited by 24 (15 self)
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theory, and change of base for groupoids and multiple
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 19 (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.
Crossed Complexes And Homotopy Groupoids As Non Commutative Tools For Higher Dimensional LocalToGlobal Problems
"... We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections. ..."
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Cited by 18 (7 self)
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We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections.
Drinfel'd Doubles And Ehresmann Doubles For Lie Algebroids And Lie Bialgebroids
, 1998
"... 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 purposes by ..."
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Cited by 13 (1 self)
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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 purposes by Liu, Weinstein and Xu. This is achieved in terms of an abstract notion of double Lie algebroid (where "double" is now used in the Ehresmann sense) which unifies many iterated constructions in differential geometry.
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 13 (10 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
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 9 (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.
Tensor categories attached to double groupoids
 Adv. Math
, 2006
"... Abstract. The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal grouplike element and q ..."
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Cited by 5 (2 self)
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Abstract. The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal grouplike element and quantum and FrobeniusPerron dimensions of simple objects are computed. Contents
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 5 (5 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.