Questions tagged [lie-groupoids]

A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group.

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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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Why are gerbes so scarce in recent research?

I have recently been exposed to gerbes for an undergraduate research project and I feel like this structure has very little mention anywhere really. Most papers that deal with anything relating to it ...
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Is the product of two Lie groupoids always a Lie groupoid?

Let $\mathcal{G}=[G_1 \rightrightarrows G_0]$ and $\mathcal{H}=[H_1 \rightrightarrows H_0]$ be two Lie groupoids. Consider the product category $\mathcal{G} \times \mathcal{H}$. My question: Is the ...
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How units of a Lie groupoid act on a manifold?

The definition of a Lie groupoid action given in Sébastien Racanière's notes (up to notation) says that the action of a Lie groupoid $\mathcal{G} \rightrightarrows M$ on a smooth manifold $Q$ consists ...
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46 views

Lie groupoid associated to an incomplete vector field

Let $M$ be a smooth manifold and $X \in \mathfrak{X}(M)$ be a vector field. If $X$ is complete, its flow defines a group action $\Bbb R \circlearrowright M$ via $t \cdot x \doteq \Phi_X(t,x)$. This ...
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76 views

Cancellation law in a Lie groupoid

I'm playing around with the definition of a Lie groupoid following Eckhard Meinrenken's notes. I read the thing for the first time in my life like an hour ago, so assume that I don't know anything ...
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50 views

Is the product of source and target maps of a Lie Groupoid a submersion?

Let $G:= (G_1 \rightrightarrows G_0)$ be a Lie Groupoid. By definition, we know that source $s$ and target $t$ are submersion. Now define $(s,t):G_1 \rightarrow G_0 \times G_0$ as $\gamma \mapsto (s(\...
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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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102 views

Quotient by a topological groupoid.

Let $G$ be a group acting on a topological space $X$, then the quotient map $X \to X/G$ is open. I want to ask, whether this fact generalizes to orbit spaces of groupoids. More precisely: Let $G$ be a ...
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How to realize the map $\eta$ globally?

I have a given map $\Phi: \mathcal{G}\longrightarrow \mathcal{H}$ between two groupoids such that $\Phi_g: \mathcal{G}_x\longrightarrow \mathcal{G}_y$ is a functor between the groupoids $\mathcal{G}_x$...
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37 views

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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Equivalence of Lie groupoids $\phi: H \rightarrow G$ induces an equivalence of categories $\phi^*: G\text{-spaces} \rightarrow H\text{-spaces}$.

In Orbifolds as Groupoids there is the notion of an equivalence $\phi: H \rightarrow G$ between Lie groupoids (2.4) and of $G$-spaces (5.1). Given a smooth functor $\phi: H \rightarrow G$ we can ...
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139 views

pull back of Groupoid spaces

I am reading Orbifolds as Groupoids: an Introduction https://arxiv.org/abs/math/0203100 Let $\mathcal{G}$ be a Lie groupoid. A right $\mathcal{G}$ space is a smoooth manifold $E$ equipped with an ...
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222 views

Group action map of a Lie group on a smooth manifold being submersion

Let $G$ be a Lie group action on a manifold $M$. It is assumed that the action is smooth i.e., $G\times M\rightarrow M$ given by $(g,m)\mapsto gm$ is a smooth map. I am trying to understand in what ...
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157 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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Gleason Yamabe for groupoids

A colleague of mine seems convinced that there is a Gleason-Yamabe type theorem for locally compact groupoids. Does anyone know if this is true? If so, any references would be most appreciated.
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On linear representations of Lie groupoids?

First, some notations and definitions: 1) For a vector space $V$:$$\mathsf{End}(V):=\{\mathsf{Linear}\ \mathsf{maps}\ f:V\longrightarrow V\}.$$ 2) A linear representation of a group $G$ is a pair $(V,...