Questions tagged [lie-algebroids]

In differential geometry, Lie algebroids generalize on one hand Lie algebras, on the other hand the tangent bundle of a manifold: they are vector bundles equipped with an anchor map, i.e. a vector bundle morphism to the tangent bundle, and a Lie algebra structure on the space of sections subject to certain Leibniz rules. The integrated version of a Lie algebroid is a Lie groupoid. A purely algebraic version is a Lie-Rinehart algebra.

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Connection between Grothendieck's homotopy hypothesis and Lie's second and third theorems?

I'm not an expert on homotopy theory, but I speculated about this in my thesis, so I figured I'd ask about it here. As I understand it, the homotopy hypothesis says that $\infty$-groupoids, with $\...
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2 votes
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Lie algebroid in algebraic geometry

When I did net-surfing at home, I met some geometric backgrounds of Lie algebras and encountered the concept of Lie algebroids. In differential geometry, a Lie algebroid seems to be defined as ...
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Is there a classifying space for transitive Lie algebroids? If so, what is it?

Let $M$ be a manifold. The data of a Lie groupoid over $M$ is equivalent to the data of a singular foliation $M=\sqcup\mathcal{F}_i$ and, for each $i$, a map (mod homotopy) $f_i:F_i\to BG_i$ (where $...
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3 votes
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How to think about representation upto homotopy of Lie algebroids

This is about the notion of representations upto homotopy of Lie algebroids. I am following the reference Representations up to homotopy of Lie algebroids by Camilo Arias Abad and Marius Crainic. Let $...
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2 votes
1 answer
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Lie algebroid associated to a vector bundle

Let $E\rightarrow M$ be a vector bundle. Kirill Mackenzie in the book General theory of Lie groupoids and Lie algebroids associates a Lie algebroid to $E\rightarrow M$ in the following steps: talk ...
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3 votes
2 answers
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Use of theory of Lie algebroids in (better) understanding of generalised complex structures

Let $M$ be a smooth manifold. A Lie algebroid over $M$ is a vector bundle $E\rightarrow M$ over $M$, with a Lie bracket on $\Gamma(M,E)$, a morphism of vector bundles $\rho:E\rightarrow TM$, such that,...
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2 votes
1 answer
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Special cases of Lie II for groupoids using elementary techniques

I asked a similar question on math.stackexchange but did not get any responses, so I thought I'd kick it up to mathoverflow. In Crainic and Fernandes's "Integrability of Lie Brackets" (and ...
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Fibration on the category of Lie pseudoalgebras implementing comorphisms

I am trying to understand comorphisms of Lie pseudoalgebras from the point of view of fibred categories, but failing miserably so far. My question would be: Is there a (op)fibration $\mathrm{LiePs} \...
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3 votes
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Natural appeareances of (commutative) algebras in $\mathfrak g$-modules

$\newcommand{\g}{\mathfrak g}$ Let $\g$ be a Lie algebra, and observe that since $U(\g)$ is a cocommutative Hopf algebra, it makes sense to look for (naturally arising and perhaps commutative?) ...
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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 Rui Loja Fernandes, ...
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Generalisation of the notion of operad

Let $\mathscr P$ be an operad in the category of vector spaces. An algebra (of the type encoded by $\mathscr P$) on the vector space $V$ is a morphism of operads $\mu:\mathscr P\to End_V$ with $End_V$ ...
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Example of tensor category with non-simple unit $J\to \mathbb{1} \to Q$ and suitably extension $Q\to M\to J$

Edit: Thanx very much to Neil Strickland for quickly explaining to us that the following cannot be realized over finite commutative $\mathbb{C}$-algebras, as I had originally asked. I know that there ...
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6 votes
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The connection between Lie algebroids and foliations

I need a bit of clarification about some of the geometry underlying the connection between Lie algebroids and foliations. In case of any confusion I'm using the definition of Lie algebroid from here. ...
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3 votes
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Is there a precisely formulable obstruction for the tangent bundle being a Lie algebra bundle?

Although vector fields (which are sections of the tangent bundle) form Lie algebras, the bundle itself, as far as I know, almost never carries Lie algebra structure; that is, in general, I believe, ...
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Computation with Lie algebroid differential?

Let $\Phi:A_M\longrightarrow A_N$ be a morphism of Lie algebroids covering $\phi:M\longrightarrow N$. Suppose $\Theta$ is a section of the pullback bundle $\phi^* A_N$. How to compute $$\langle d_{...
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Lie Groupoid of a Transitive Lie algebroid

If A is an integrable and transitive Lie Algebroid, and G is a corresponding Lie groupoid, then: is G necessarily transitive too? I guess it is not generally true, but I wonder under which conditions ...
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Is $\mathbb{P}^1$ the only smooth projective curve with a locally split tangent lie algebroid?

Let $C$ be a smooth projective curve over an algebraically closed field $k$. The tangent lie algeborid $\mathcal{T}_C$ of $C$ is just sheaf of vector fields on $C$ equipped with the usual lie ...
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16 votes
2 answers
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Is every singular foliation induced by a Lie algebroid?

Let $M$ be a smooth manifold. A smooth distribution $D$ on $M$ is the union of a family $\{D_p \leq T_p M : p\in M\}$ of vector spaces such that there is a family $\mathcal C $ of smooth vector ...
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Lie Algebroid Structure on $A_M\times I\longrightarrow M\times I$?

Let $p_{A_M}:A_M\longrightarrow M$ be a Lie algebroid and $I:=[0, 1]$. Then $$p_{A_M}\times \textrm{id}:A_M\times I\longrightarrow M\times I,$$ is a vector bundle. There is a $C^\infty(M\times I)$-...
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Analogue of Kontsevich's formality theorem for quantization of Courant algebroids

In his 1997 preprint, M. Kontsevich proved the formality of the differential graded algebra controlling deformations of the associative and commutative algebra of functions on a manifold, seen as an ...
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141 views

Action of a Lie groupoid on a Lie Algebroid?

Let $\pi:E\longrightarrow M$ be a vector bundle. Then we can associate a Lie groupoid $\mathsf{Gl}(E)\rightrightarrows M$ where $$\mathsf{Gl}(E):=\{E_x\stackrel{lin. isom.}{\longrightarrow} E_y: x, y\...
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Property of Lie Algebroid Morphism?

Let $A\longrightarrow M$ and $B\longrightarrow N$ be two Lie algebroides and $\Phi:A\longrightarrow B$ a morphism of Lie algebroids covering $\phi:M\longrightarrow N$. Let $\alpha, \beta\in \Gamma(A)$....
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3 votes
1 answer
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Poisson structure on the dual Lie algebroid

Let $E \to X$ be a Lie algebroid over the manifold $X$. Let $x_1,...x_n$ be local coordinates on $X$ and $e_1,...e_m$ be the basis of local sections of $E$. In terms of these coordinate functions Lie ...
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1 vote
1 answer
175 views

Property of Lie algebroid morphism: $\#_B\circ \Phi=d\phi\circ \#_A$?

Let $A\longrightarrow M$ and $B\longrightarrow N$ be Lie algebroids with anchors $\#_A$ and $\#_B$, respectively. A morphism of Lie algebroids is a morphism of vector bundles $\Phi:A\longrightarrow ...
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4 votes
1 answer
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A-Paths as morphisms of Lie Algebroids $TI\longrightarrow A$?

In the paper Integrability of Lie Brackets Marius Crainic and Rui Fernandes describe obstructions to integrate a Lie algebroid to a Lie groupoid. The process of integration relies on the construction ...
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9 votes
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434 views

Differentiation of Lie $\infty$-groupoids

I've been trying to understand how to differentiate Lie $\infty$-groupoids to get a Lie $\infty$-algebroid. First of all, I will state the definitions that I'm assuming. A Lie $\infty$-groupoid is a ...
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10 votes
1 answer
424 views

Almost but not quite a Lie algebroid: what is it?

In some calculations, I have arrived at the following algebraic structure, reminiscent of a Lie algebroid, but not quite. I have a real line bundle $E \to M$, on whose smooth sections $\Gamma(E)$ I ...
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5 votes
1 answer
654 views

AKSZ sigma models for higher spin

The AKSZ framework constructs 2D sigma models in the BV formalism. Is there a generalization of the AKSZ approach to higher spin?
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6 votes
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Continuous and smooth Lie groupoid cohomology

In the paper by Weinstein and Xu: Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), there are two versions of Lie groupoid cohomology. The same differential $\...
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2 votes
1 answer
472 views

What's an example of a commutative algebra over $\mathbb Q$ that fails to satisfy this version of the "PBW theorem"

In a recent question, I recalled the notion of differential operator, polyderivation, and principal symbol for a commutative algebra $A$ over some fixed commutative ring $k$. (I will not repeat those ...
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9 votes
1 answer
629 views

For which algebras does \{Differential Operators\} satisfy a PBW-like theorem?

Let $k$ be a commutative ring, $A$ a commutative $k$-algebra, and for some other part of why I'm asking this question I only care about the case when $k \supseteq \mathbb Q$. Recall the following ...
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12 votes
2 answers
2k views

Differential forms on an almost complex manifold

Hello! Let $M$ be an almost complex manifold. Let $TM$ denote its tangent bundle. Then we have the decomposition $TM\otimes\mathbb{C}=T^{1,0}M\oplus T^{0,1}M$ corresponding to the eigenvalues of the ...
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3 votes
3 answers
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Examples of Lie Algebroids

The concept of a Lie Algebroid is given an important geometric meaning in the framework of Generalized Complex Geometry. For reference, the (barebones) definition of a Lie Algebroid is a vector bundle ...
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3 votes
3 answers
844 views

Is there any relation between deformation and extension of Lie algebras?

In a paper of A. Weinstein on the geometry of Poisson manifolds, he relates the formal linearization around a zero, p, of the Poisson bivector to extensions of the Lie algebra induced by the bivector ...
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5 votes
3 answers
883 views

Geometry and Integrability in Other Bundles

Background: Suppose $E=TM$ is the tangent bundle to some differentiable manifold $M^n$. If we specify some subbundle $D\subset TM$ (distribution of $k$-planes) then there are two natural situations ...
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2 votes
2 answers
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Is the cohomology of the corresponding Lie algebroid an invariant under equivalence of source-simply-connected Lie groupoids?

Recall the related notions of Lie groupoid, Lie algebroid, generalized morphism of Lie groupoids, and cohomology of Lie algebroid. Henceforth, I will drop the word "Lie" for all those things listed ...
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3 votes
1 answer
308 views

When does a VBLA induce an isomorphism on Lie algebroid cohomology?

This question is geared towards the experts, so I will only briefly gloss the definitions. Everything I say is in the category of finite-dimensional smooth manifolds, and whenever I say "$\mathbb Z$-...
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5 votes
1 answer
902 views

Do Lie algebroids pull back (along submersions)?

There are more general definitions, but for my purposes a Lie algebroid on a smooth manifold $X$ is a vector bundle $A \to X$, a map $\rho: A \to {\rm T}X$ of vector bundles over $X$, and a bracket $[,...
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3 votes
3 answers
560 views

What is an obviously coordinate-independent description of the Chevellay-Eilenberg complex for a Lie algebroid?

I've read in many places, including the n-Lab page, that a Lie algebroid (which I think of as in the first definition on the n-Lab page) is the same as a vector bundle $A \to X$ and a (properties?) ...
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