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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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First examples of Lie-Rinehart algebras that are not coming from Lie algebroids

I heard the idea of a Lie-Rinehart algebra first time from an algebraist. I noticed there is a similarity between description of Lie algebroid on a manifold and the algebraic notion of Lie-Rinehart ...
Praphulla Koushik's user avatar
5 votes
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Does the forgetful functor from Lie algebroids to vector bundles have a right adjoint

Let $\mathcal{S}$ be an appropriate category of smooth spaces, and let $\mathrm{Vect} \colon \mathcal{S}^\mathrm{op} \to \mathsf{Cat}$ be the functor sending a space to the category of vector bundles ...
cheshircat's user avatar
1 vote
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graded cocommutative and coassociative coalgebra, cofree in the category of locally nilpotent differential graded coalgebras

For someone who is new to Lie $\infty$-algebras, the title looks confusing. This is how Lie $\infty$-algebras are commonly described, for example, see What is a homotopy between $L_\infty$-algebra ...
Praphulla Koushik's user avatar
2 votes
1 answer
224 views

Regarding first order differential operator and derivative endomorphism

I am reading "Differential operators and actions of Lie algebroids" by Kosmann-Schwarzbach and Mackenzie. There is some confusion regarding the terminology. Let $E\rightarrow M$ be a vector ...
Praphulla Koushik's user avatar
2 votes
0 answers
177 views

Cohomology theory for Dirac manifolds

I am trying to see if there is any existing cohomology theory for Dirac manifolds. For the case of poisson manifolds, we have the notion of Poisson cohomology. For a manifold $M$, one can consider the ...
Praphulla Koushik's user avatar
4 votes
0 answers
118 views

Averaging over a Weinstein groupoid?

(Not sure if this question belongs here or on m.SE) For a Lie group, $G$ (of dimension $n$), one can average over the group: $$ \Gamma = \int_{G} d\mu(g) ~g $$ (where $d\mu(g)$ is the left-Haar ...
dennis's user avatar
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Is there a notion of representation theory of foliations?

A foliation on a manifold $M$ can be seen as a sub bundle of the tangent bundle $\mathcal{F}\subseteq TM$ that is closed under Lie bracket of vector fields. One can think of foliation as a Lie ...
Praphulla Koushik's user avatar
2 votes
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183 views

Relation between equivariant geometry and representation theory (of geometric objects)

Equivariant geometry studies "manifolds" with an extra structure $G\times M\rightarrow M$. Representation theory studies "Lie algebroids" with an extra structure $\Gamma(M,A)\times ...
Praphulla Koushik's user avatar
0 votes
0 answers
60 views

Jacobi identity for the bracket in action Lie algebroid

I am trying to verify that the bracket in the definition of action Lie algebroid satisfies the Jaboci Identity. Let $X \to X^\dagger$ be an action of a Lie algebra $\mathfrak{g}$ on a manifold $M$, i....
Saikat's user avatar
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1 answer
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references to learn the general theory Lie $\infty$-groupoids and Lie $\infty$-algebroids

Kirill Mackenzie has a book on the general theory of Lie groupoids and Lie algebroids. Is there such a reference for the general theory of Lie $\infty$-groupoids and Lie $\infty$-algebroids; that ...
Praphulla Koushik's user avatar
8 votes
1 answer
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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 $\...
Josh Lackman's user avatar
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2 votes
1 answer
343 views

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 ...
Frank Voigt's user avatar
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108 views

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 $...
Doron Grossman-Naples's user avatar
3 votes
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161 views

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 $...
Praphulla Koushik's user avatar
3 votes
2 answers
523 views

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 ...
Praphulla Koushik's user avatar
3 votes
2 answers
336 views

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,...
Praphulla Koushik's user avatar
2 votes
1 answer
116 views

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 ...
Ben MacAdam's user avatar
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7 votes
1 answer
320 views

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} \...
Marvin Dippell's user avatar
3 votes
0 answers
55 views

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?) ...
Pedro's user avatar
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3 votes
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248 views

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, ...
Murilo Luiz's user avatar
3 votes
0 answers
171 views

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 $End_V$ ...
thingsthatmighthavebeen's user avatar
1 vote
1 answer
160 views

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 ...
Simon Lentner's user avatar
6 votes
0 answers
449 views

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. ...
Joe Pollard's user avatar
3 votes
0 answers
136 views

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, ...
მამუკა ჯიბლაძე's user avatar
1 vote
0 answers
57 views

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_{...
PtF's user avatar
  • 383
3 votes
1 answer
194 views

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 ...
JCPT's user avatar
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-1 votes
1 answer
294 views

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 ...
Saal Hardali's user avatar
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16 votes
3 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 fields ...
Ervin's user avatar
  • 395
2 votes
0 answers
61 views

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 $C^\infty(M\times I)$-...
PtF's user avatar
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5 votes
0 answers
195 views

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 ...
thingsthatmighthavebeen's user avatar
2 votes
0 answers
163 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\...
PtF's user avatar
  • 383
2 votes
0 answers
429 views

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 $\alpha, \beta\in \Gamma(A)$....
PtF's user avatar
  • 383
3 votes
1 answer
602 views

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 ...
cll's user avatar
  • 2,295
1 vote
1 answer
204 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 ...
PtF's user avatar
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4 votes
1 answer
276 views

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 ...
PtF's user avatar
  • 383
10 votes
0 answers
523 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 ...
user40276's user avatar
  • 2,209
10 votes
1 answer
477 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 ...
José Figueroa-O'Farrill's user avatar
5 votes
1 answer
663 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?
Jim Stasheff's user avatar
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6 votes
1 answer
589 views

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 $\...
Nicola Ciccoli's user avatar
2 votes
1 answer
496 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 ...
Theo Johnson-Freyd's user avatar
9 votes
1 answer
674 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 ...
Theo Johnson-Freyd's user avatar
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 ...
Benjamin's user avatar
  • 987
4 votes
3 answers
1k views

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 ...
Tarun Chitra's user avatar
3 votes
3 answers
858 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 ...
Feri's user avatar
  • 65
5 votes
3 answers
1k 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 ...
Justin Curry's user avatar
  • 2,694
2 votes
2 answers
370 views

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 ...
Theo Johnson-Freyd's user avatar
3 votes
1 answer
341 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 "$\mathbb Z$-...
Theo Johnson-Freyd's user avatar
5 votes
1 answer
1k 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 $[,...
Theo Johnson-Freyd's user avatar
3 votes
3 answers
620 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?) ...
Theo Johnson-Freyd's user avatar