Global analysis, analysis on manifolds Groupoids, pseudogroups and general structures Theorem

Crainic–Fernandes integrability theorem

Statement

Let $A \to M$ be a Lie algebroid with anchor $\rho \colon A \to TM$ . For $x \in M$ , let $\mathfrak{g}_x = \ker(\rho_x)$ be the isotropy Lie algebra at $x$ , with centre $Z(\mathfrak{g}_x)$ . Let $\mathcal{N}_x(A) \subseteq Z(\mathfrak{g}_x)$ be the monodromy group at $x$ . It is the set of $v \in Z(\mathfrak{g}_x)$ such that the constant $A$ -path with value $v$ is $A$ -homotopic to the zero $A$ -path at $x$ . Fix a continuous norm on the fibres of $A$ , and let $r(x)$ be the distance from $0$ to $\mathcal{N}_x(A) \setminus \set{0}$ , with $r(x) = \infty$ when $\mathcal{N}_x(A) = \set{0}$ . Then $A$ is isomorphic to the Lie algebroid of some Lie groupoid if and only if, for every $x \in M$ ,

$\mathcal{N}_x(A)$ is a discrete subset of $A_x$ , and
$\liminf_{y \to x} r(y) > 0$ .

Remarks

Crainic and Fernandes build a candidate groupoid $\mathcal{G}(A)$ , the Weinstein groupoid, as the quotient of the space of $A$ -paths by $A$ -homotopy. It is always a topological groupoid, and $A$ is integrable exactly when $\mathcal{G}(A)$ is smooth; it is then the unique integration whose source fibres are simply connected. The monodromy groups measure the failure of smoothness, and the two conditions say that they stay uniformly discrete as the point moves. The condition does not depend on the chosen norm.

Example: when $M$ is a point, $A$ is a finite-dimensional Lie algebra and the monodromy groups are trivial. So every Lie algebra integrates to a Lie group, which is Lie's third theorem. Unlike Lie algebras, Lie algebroids need not be integrable, and the theorem identifies the obstruction.

Going deeper

For a student

The theorem settles the integrability problem for Lie algebroids, which asks for the analogue of Lie's third theorem. It applies to Poisson manifolds through their cotangent Lie algebroids, where integrability means the existence of a symplectic groupoid. The monodromy group at $x$ is the image of a homomorphism defined on the second homotopy group of the orbit through $x$ . So integrability is automatic when every orbit has trivial second homotopy group.

Reading path

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Lie groupoid concept
Vector space definition
Norm definition
Crainic–Fernandes integrability theorem statement

Tag	`5ZNN348M6X`
URI	`m2://5ZNN348M6X`
Kind	statement, theorem
Writing	W0 stub
MSC	58H05, 22A22, 53D17
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About	Lie groupoid
Uses	Lie groupoid, Norm