Questions tagged [lie-algebroids]

In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones.

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Reference request: Integration of A-paths.

Recall that an A-path for a Lie Algebroid is an an algebroid morphism from the tangent bundle over the unit interval $[0,1]= I$ to a general Lie algebroid $A$. Now, whenever A is the algebroid of some ...
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Local trivializations and transition law of Atiyah Lie algebroid

Let $P\rightarrow M$ be a $G$-principal bundle and denote by $\mathrm{at}(P):=TP/G$ the Atiyah Lie algebroid over $M$. I want to understand how $\mathrm{at}(P)$ is a locally trivial Lie algebroid with ...
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Is there any reference treating explicitly Lie-Rinehart pairs over non-commutative base algebras?

Wondering around in the literature, any reference to Lie-Rinehart algebras define them as pairs $(A,L)$ where $A$ is a commutative algebra over some field $\Bbbk$ (or even commutative ring) and $L$ is ...
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Example of action of a Lie algebroid

I am studying the notions of Lie groupoids and algebroids. I am mainly interested in the specific case of Lie algebroids which are actually Lie algebra bundles over an open subset of $\mathbb{R}^k$. ...
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Integrability of Lie algebroids

In the article https://arxiv.org/pdf/math/0611259.pdf, it is defined the integrability of a Lie algebroid as follows: a Lie algebroid $A$ is integrable iff it is isomorphic to the Lie algebroid of a ...
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What are the prerequisites to learning about Lie groupoids, Lie algebroids and holomorphic foliations?

I am a graduate student of Theoretical Physics and intend to take a course titled "Introduction to Lie groupoids, Lie algebroids and holomorphic foliations". The course page doesn't have information ...
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Recovering the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids

How can I recover the Weinstein Splitting Theorem for Poisson manifolds using the Local Splitting theorem for Lie algebroids? I am using the formulation of the theorem given in https://arxiv.org/pdf/...
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Homotopy of algebroid paths

In Lectures on Integrability of Lie Brackets, Proposition 3.15, Fernandes and Crainic motivate what it means to say that two paths on an algebroid are homotopic. I spent quite some time trying to ...
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Decomposition of section pushforward

In Lectures on Integrability of Lie Brackets, while defining the concept of a non-base preserving Lie algebroid, Crainic and Fernandes claim the following (beginning of page 27): Given vector bundles ...
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Purpose of the anchor map, Lie algebroid

Let $X$ be a scheme. A Lie algebroid $L$ on $X$ is a quasi-coherent $\mathcal{O}_X$-module equipped with a morphism of $\mathcal{O}_X$-modules $\sigma:L\to \mathcal{T}_X$ the tangent sheaf, and a Lie ...
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154 views

What is the tangent Lie algebroid to a Lie groupoid?

How do you define the tangent Lie algebroid to a Lie groupoid? In this online note Lie Algebroids, Lie Groupoids and Poisson Geometry by Sébastien Racanière, it states that if $t\colon G_1\to G_0$ is ...
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What kind of structure on the tangent bundle is determined by the commutator of vector fields? Or, trying to understand algebroids.

I vaguely remember seeing this accurately described somewhere but cannot recall where, so mainly this is a reference request. Let $T$ be the tangent bundle of a smooth manifold or of an algebraic ...
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Why aren't there any derivations of degree inferior to $-1$ of the DG-algebra $(\Omega(A), d_A, \wedge)$?

Let $A$ be a vector bundle over a manifold $M$. We can assotiate a graded algebra $(\Omega(A), \wedge)$ where $$\wedge:\Omega^i(A)\times \Omega^j(A)\longrightarrow \Omega^{i+j}(A),$$ is given by $$(\...
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Pullback Courant algebroid

I am reading the lecture notes on generalized geometry http://www.staff.science.uu.nl/~caval101/homepage/Research_files/australia.pdf but now I am stuck on the exercise 1.31 at page 13. Here is the ...
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Lie Algebroids as Sheaves?

In these notes the author defines a Lie algebroid as follows: A Lie algebroid on $X$ is a sheaf of vector spaces $\mathcal{A}$ together with: $(a)$ a structure of $\mathcal{O}_X$-module on $\mathcal{...
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A-paths on Lie algebroids?

In the paper integrability of Lie brackets M. Crainic and L. R. Fernandez define the notion of A-path as follows. Definition. Let $A\stackrel{\pi}{\longrightarrow} M$ be a Lie algebroid. An A-path is ...