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Topic: Lie algebroid

	Lie algebroid - Wikipedia, the free encyclopedia
		In mathematics, a Lie algebroid is a triple (E,[,],ρ) consisting of a vector bundle E over a manifold M, togehter with a Lie bracket [,] on its module of sections Γ(E) and a morphism of vector bundles
		1) To every Lie groupoid is associated a Lie algebroid analogous to the correspondence of a Lie algebra to a Lie group.
		2) Given the action of a Lie Algebra g on a Manifold M, the set of g -invariant vector fields on M is a Lie Algebroid over the space of orbits of the action.
	en.wikipedia.org /wiki/Lie_algebroid (155 words)

	Abstracts (Site not responding. Last check: 2007-10-09)
		A differential groupoid is a groupoid whose space of morphisms and space of units are differentiable manifolds (possibly with corners) such that all structural maps are differentiable and the domain map is a submersion.
		To a differential groupoid one can associate a Lie algebroid, which is a vector bundle whose sections identify with the right invariant vector fields on the groupoid that are tangent to the fibers of the domain map.
		A main result is that a Lie algebroid on a stratified manifold is integrable if, and only if, its restriction to each strata is integrable and the exponential map is an isomorphism in a neighborhood of the zero section.
	www.math.psu.edu /nistor/abstracts.html (2189 words)

	Lie Algebra -- from Wolfram MathWorld
		A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket.
		The classification of finite dimensional simple Lie algebras over an algebraically closed field of characteristic 0 can be accomplished by (1) determining matrices called Cartan matrices corresponding to indecomposable simple systems of roots and (2) determining the simple algebras associated with these matrices (Jacobson 1979, p.
		This is one of the major results in Lie algebra theory, and is frequently accomplished with the aid of diagrams called Dynkin diagrams.
	mathworld.wolfram.com /LieAlgebra.html (209 words)

	Lie-Poisson bracket (Site not responding. Last check: 2007-10-09)
		Given any Lie algebroid, the image of the anchor is always a generalized integrable distribution; its maximal integrable submanifolds are called orbits of the Lie algebroid.
		is a Lie algebroid with anchor the sharp map.
		The orbits of this algebroid are the symplectic leaves of
	www.mimuw.edu.pl /~pwit/TOK/sem4/online/node8.html (58 words)

	Drinfel'd Doubles And Ehresmann Doubles For Lie Algebroids And Lie Bialgebroids (ResearchIndex)
		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...
		26 Lie groupoids and Lie algebroids in differential geometry (context) - Mackenzie - 1987
	citeseer.ist.psu.edu /mackenzie98drinfeld.html (562 words)

	AMCA: The fundamental form on a Lie groupoid of diffeomorphisms by Ivan Belko
		It is based on the theory of jets and Lie groupoids and it is the development of E.
		(B) consists of k-jets of local diffeomorphism of B. The Lie groupoid \Pi (B) is a Lie groupoid of linear isomorphisms of the tangent bundle TB.
		This Lie algebroid is isomorphic to Lie algebroid J
	at.yorku.ca /c/a/k/m/18.htm (426 words)

	"Pontryagin algebra of a transitive Lie algebroid" (Site not responding. Last check: 2007-10-09)
		On the other hand, in analogy to the theory of Lie groups and Lie algebras, each principal fibre bundle P has its algebraic equivalent: a transitive Lie algebroid A(P) - constructed on the basis of the right-invarint vector fields on P.
		We pay our attention to the fact that this holds although in the Lie algebroid A(P) there is no direct information about the structural Lie group of P which can be disconnected.
		In connection with the above it is important to construct the Chern-Weil homomorphism of a transitive Lie algebroid.
	im0.p.lodz.pl /~kubarski/forum/abs14.html (270 words)

	Problems on groupoids (Site not responding. Last check: 2007-10-09)
		is the Lie algebroid of a Lie groupoid G.
		The construction of the tangent groupoid has been extended to local Lie groupoids in the context of pseudo-differential operators by Nistor, Weinstein and Xu.
		In fact, I had to integrate a Lie algebroid map to a groupoid map in my work.
	unr.edu /homepage/ramazan/groupoid/open_prb/gop.html (629 words)

	Atlas: Lie-Rinehart algebras, descent, and quantization by Johannes Huebschmann (Site not responding. Last check: 2007-10-09)
		The quantization problem for a Poisson algebra is related to that of constructing suitable representations of a Lie-Rinehart algebra associated with the Poisson algebra in a natural fashion, as explained in our paper entitled ``Poisson cohomology and quantization'', J. für die reine und angewandte Mathematik 408 (1990), 57-113.
		Examples of stratified Kähler spaces arise from the closures of holomorphic nilpotent orbits in a semisimple Lie algebra of hermitian type, including angular momentum zero reduced spaces; in this case, the Lie-Rinehart structure on the reduced level admits a direct description in terms of the Lie structure of the Lie algebra of hermitian type.
		Other examples of stratified Kähler spaces arise from representations of compact Lie groups as well as from spaces of possibly twisted representations of the fundamental group of a surface in a compact Lie group.
	atlas-conferences.com /cgi-bin/abstract/cajf-28 (339 words)

	Grabowski, Marmo, Michor: Homology and modular classes of Lie algebroids
		For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory.
		Fernandes, “Lie algebroids, holonomy and characteristic classes”, Adv.
		Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Cambridge University Press, 1987 MR 896907
	aif.cedram.org /aif-bin/item?id=AIF_2006__56_1_69_0 (269 words)

	Transactions of the American Mathematical Society
		Abstract: Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups.
		J. Kubarski, Lie algebroid of a principal fibre bundle, Publ.
		------, A criterion for the minimal closedness of the Lie subalgebra corresponding to a connected nonclosed Lie subgroup, Rev. Mat.
	www.ams.org /tran/1996-348-06/S0002-9947-96-01646-7 (332 words)

	Lie Algebroid -- from Wolfram MathWorld
		The infinitesimal algebraic object associated with a Lie groupoid.
		Lie bracket) on its space of smooth sections together with its anchor
		Weisstein, Eric W. "Lie Algebroid." From MathWorld--A Wolfram Web Resource.
	mathworld.wolfram.com /LieAlgebroid.html (76 words)

	Integration of twisted Dirac brackets, Henrique Bursztyn, Marius Crainic, Alan Weinstein, Chenchang Zhu
		Given a Lie groupoid G over a manifold M, we show that multiplicative 2-forms on G relatively closed with respect to a closed 3-form ϕ; on M correspond to maps from the Lie algebroid of G into T
		M satisfying an algebraic condition and a differential condition with respect to the ϕ-twisted Courant bracket.
		[17] P. Higgins and K. Mackenzie, Algebraic constructions in the category of Lie algebroids, J. Algebra 129 (1990), 194--230.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.dmj/1086957716 (673 words)

	DC MetaData for: Lie Algebroid Morphisms, Poisson Sigma Models, and Off-Shell Closed Gauge Symmetries (Site not responding. Last check: 2007-10-09)
		Lie Algebroid Morphisms, Poisson Sigma Models, and Off-Shell Closed Gauge Symmetries
		field equations are interpreted as morphisms of Lie algebroids and their
		Yang-Mills type gauge theories of Lie algebroids, which include the standard
	www.esi.ac.at /Preprint-shadows/esi1490.html (155 words)

	AMCA: Quadartization of Lie algebroid by Michel Nguiffo Boyom (Site not responding. Last check: 2007-10-09)
		AMCA: Quadartization of Lie algebroid by Michel Nguiffo Boyom
		Then, the linear part of the Taylor expansion at 0 of \Pi defnes a Lie algebra structure on the vector space of linear functions on R
		According to Alan WEINSTEIN, a Lie algebra L is formally (resp.
	at.yorku.ca /c/a/k/m/22.htm (193 words)

	Lie algebroid - Education - Information - Educational Resources - Encyclopedia - Music (via CobWeb/3.1 ... (Site not responding. Last check: 2007-10-09)
		Lie algebroid - Education - Information - Educational Resources - Encyclopedia - Music (via CobWeb/3.1 planetlab1.netlab.uky.edu)
		can be thought of as a restricted Lie module that has both a Lie bracket and a Lie algebra morphism, known as an anchor map, given as
		The standard example of a Lie algebroid is the identity map corresponding to the tangent space of a Lie groupoid.
	education.music.us.cob-web.org:8888 /L/Lie-algebroid.htm (297 words)

	Integrability of Lie brackets, Marius Crainic, Rui Loja Fernandes (Site not responding. Last check: 2007-10-09)
		In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids.
		We show that the integrability problem is controlled by two computable obstructions.
		As applications we derive, explain and improve the known integrability results, we establish integrability by local Lie groupoids, we clarify the smoothness of the Poisson sigma-model for Poisson manifolds, and we describe other geometrical applications.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.annm/1052148097 (76 words)

	Copyright Permission - Differential Geometry and its Applications - On transitive Lie bialgebroids and Poisson groupoids
		Copyright.com is the place to go for the right to use and share content from millions of print and online information sources.
		Copyright (c) 2006 Elsevier B.V. We prove that, for any transitive Lie bialgebroid (A, A^), the differential associated to the Lie* algebroid structure on A^* has the form d
		+Ω, where Λ is a section of ±?^2A and Ω is a Lie algebroid 1-cocycle for the adjoin...
	www.copyright.com /articles/sd/7a2/09262245/article_09262245_1304.html (197 words)

	Math arXiv: Search results (Site not responding. Last check: 2007-10-09)
		math.DG/0312162 Derivations of the Lie Algebras of Differential Operators.
		math.DG/0203053 The Lie algebra of a Lie algebroid.
		dg-ga/9710013 Tangent and cotangent lifts and graded Lie algebras associated with Lie algebroids.
	front.math.ucdavis.edu /author/Grabowski-J* (325 words)

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