Algebraic constructions in the category of lie algebroids
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In this article, it was shown that the Lie functor from the category of all differentiable groupoids (over arbitrary bases) and arbitrary smooth morphisms, to the class of all Lie algebroids, preserves the basic algebraic constructions known to be possible in (differentiable) groupoids.About:
This article is published in Journal of Algebra.The article was published on 1990-02-15 and is currently open access. It has received 354 citations till now. The article focuses on the topics: Lie algebroid & Lie groupoid.read more
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General theory of lie groupoids and lie algebroids
TL;DR: A comprehensive modern account of the theory of Lie groupoids and Lie algebroids, and their importance in differential geometry, in particular their relations with Poisson geometry and general connection theory, is given in this article.
Journal ArticleDOI
Integrability of Lie brackets
TL;DR: In this paper, the integrability problem of Lie algebroids is solved by two computable obstructions, i.e., local Lie groupoids and the smoothness of the Poisson sigma model.
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Lie bialgebroids and Poisson groupoids
Kirill C. H. Mackenzie,Ping Xu +1 more
TL;DR: In this article, the authors introduce and study a natural infinitesimal invariant for Poisson groupoids, the Lie bialgebroids of the title, which is a special case of Lie algebras satisfying a triangularity condition.
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Integrability of Lie brackets
TL;DR: In this paper, the integrability problem of Lie algebroids is solved by two computable obstructions, i.e., local Lie groupoids and the smoothness of the Poisson sigma model.
Journal ArticleDOI
Integrability of Poisson Brackets
TL;DR: In this paper, the integration of Poisson manifolds was studied in terms of variations of symplectic areas, and the existence of complete symplectic realizations was shown to exist.
References
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Categories for the Working Mathematician
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Book
Foundations of Differentiable Manifolds and Lie Groups
TL;DR: Foundations of Differentiable Manifolds and Lie Groups as discussed by the authors provides a clear, detailed, and careful development of the basic facts on manifold theory and Lie groups, including differentiable manifolds, tensors and differentiable forms.
Book
Lie Groupoids and Lie Algebroids in Differential Geometry
TL;DR: In this article, the authors provide a unified and detailed account of the theory of Lie groupoids and Lie algebroids, and apply this theory to the cohomology of Lie groups, re-interpreting connection theory in cohomological terms, and giving criteria for the existence of (not necessarily Riemannian) connections.
Book
Treatise on Analysis
TL;DR: Weyl-Kodaira theory: elliptical differential operators on an interval of R boundary conditions self-adjoint operators associated with a linear differential equation the case of second order equations example -second order equations with periodic-coefficients example -Gelfand-Levitan equations as mentioned in this paper.