General theory of lie groupoids and lie algebroids

By K. Mackenzie. 2005

1,104 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e598fa392c09a96e519b0ca6835b143c70884e63

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Differentiable stacks and gerbes (2006, background): “sentable submersion, all Xp are manifolds. Of course, X 1 ⇉ X 0 is the Lie groupoid associated to the atlas X →X. Furthermore, we assume that X 1 ⇉ X 0 is a Hausdorﬀ and second countable Lie groupoid [28, 38]. We have p+1 projection maps Xp→Xp−1, giving rise to a diagram ... //// //// X 2 // // //X ”
Free Lie algebroids and the space of paths (2007, background): “See [Mack] for more background and examples of Lie groupoids.”
Étale groupoids and their quantales (2004, background): “, [2, 3, 5, 12, 13, 14, 17, 27, 28]), and in this paper we intend to provide evidence showing that also the theory of quantales may have a natural role to play in this context, indeed providing useful new insights and techniques.”
Linear and Multiplicative 2-Forms (2009, background): “There is a natural double vector bundle [12, 15] associated with it, referred to as the tangent double vector bundle of A, and defined by the following diagram:”
VB-groupoids and representation theory of Lie groupoids (2010, background): “Then [14, 16] there is an associated dual VB-groupoid”
Moving Frames for Lie Pseudo–Groups (2008, neutral): “[18, 40, 53].”
Twisted Differential String and Fivebrane Structures (2009, background): “• For arbitrary X and g concentrated in degree 0 (being finitely generated and projective as a module over C(X)) this is equivalent to the usual definition of Lie algebroids as vector bundles E → X with anchor map [59] ρ : E → TX: we have g = Γ(E) and the anchor is encoded as dg|C∞(X) : f 7→ ρ(·)(g)”
On the universal enveloping algebra of a Lie algebroid (2010, background): “In this section we will briefly recall the definition of Lie algebroids and mention some basic examples (see [9, 12] for more details, examples and historical remarks).”
Multiplicative forms at the infinitesimal level (2010, background): “[21]) to obtain a unique groupoid morphism”
Ehresmann doubles and Drinfel'd doubles for Lie algebroids and Lie bialgebroids (2006, methodology): “These two duals are themselves duals; see [35], Chapter 9.”
Lectures on integrability of Lie brackets (2006, background): “ Karas¨ev [34] and Zakrzewski [72])). The book of MacKenzie [37] contains a nice introduction to the theory of transitive and locally trivial groupoids, which has now been superseeded by his new book [39], where he also treats non-transitive groupoids and double structures. The modern approach to gr”
Geometric structures as deformed infinitesimal symmetries (2004, background): “3 and the work cited above for further examples.”
Calculus on Lie algebroids, Lie groupoids and Poisson manifolds (2008, background): “Formulae in local coordinates for these algebroid structures are given in [17], and other propertie of these lifts can be found in [35].”
Splitting theorems for Poisson and related structures (2016, methodology): “(E) such that the corresponding operator D on sections is a derivation of the Lie bracket. In particular, the operators Dσ= [σ,·] deﬁne inﬁnitesimal automorphisms ea(σ) ∈ autLA(E). The resulting lift (30) ea: Γ(E) → autLA(E). has the property (23). 18 HENRIQUE BURSZTYN, HUDSON LIMA, AND ECKHARD MEIN”
Graded bundles and homogeneity structures (2011, background): “imilarly, one can deﬁne the isomorphism I′ : Tm,nM →TmTnM such that I′ ◦I−1 gives a canonical isomorphism TnTmM→TmTnM. For m= n= 1 we recover the well-known involution of the double vector bundle TTM([6], p. 363). For n= 1, we have an isomorphism TTmM≃TmTM whose dual version, T∗TmM≃TmT∗Mhas been stu”