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Results tagged with lie-algebroids
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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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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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answer
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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 bundle.
A …
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graded cocommutative and coassociative coalgebra, cofree in the category of locally nilpoten...
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 mor …
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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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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 \Gamma(M,E)\rightar …
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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 algebro …
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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 ab …
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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 …
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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 al …
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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 cove …
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