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May 7, 2008

Integrability of Lie Brackets

Posted by Urs Schreiber

I would like to advertise the beautiful review

Marius Crainic, Rui Loja Fernandes
Lectures on Integrability of Lie Brackets
arXiv:math/0611259

on the integration of Lie algebroids (g,A) to Lie groupoids C(g,A).

Section 3.2 has a nice review of the method of integrating Lie algebras to Lie groups using equivalence classes of paths in the Lie algebra. Then in 3.3 it is discussed how this generalizes to Lie algebroids.

In section 5.3 of On action Lie ∞-groups and action Lie ∞-algebras (pdf) I describe how this integration method is secretly (well, it’s pretty obvious, but still deserves to be made explicit) nothing but forming the fundamental path groupoid Π 1 (−) of the smooth classifying space S(CE(g,A)) of (g,A)-valued differential forms: C(g,A)=Π 1 (S(CE(g,A))).

Posted at May 7, 2008 8:41 AM UTC

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Read the post (Action) Lie Infinity-Algebroids
Weblog: The n-Category Café
Excerpt: Talk on the Lie infinity-algebroid perspective on the BRST complex and its relation to the integrated picture of action infinity-groups.
Tracked: May 8, 2008 12:08 PM

Re: Integrability of Lie Brackets

Do these folks get a smooth groupoid from any Lie algebroid, or just a stacky Lie groupoid?

The latter approach, I claim, is just a consequence of working in an insufficiently nice category of smooth spaces.

Posted by: John Baez on May 15, 2008 7:15 PM | Permalink | Reply to this

Re: Integrability of Lie Brackets

Do these folks get a smooth groupoid from any Lie algebroid, or just a stacky Lie groupoid?

They discuss in detail the obstructions on a Lie algebroid to it integrating to a groupoid internal to manifolds.

If the obstruction vanishes, they construct the groupoid internal to manifolds. If it does not vanish, I think they don’t do anything further but just notice that the construction internal to manifolds fails.

The latter approach, I claim, is just a consequence of working in an insufficiently nice category of smooth spaces.

Yes, certainly.

Every L ∞-algebroid integrates to an ∞-groupoid internal to “SmoothSpaces”, when the latter is taken to be sheaves on Euclidean spaces. This is just abstract nonsense. With sufficient luck and sufficient labor, we can then try to see if we can move down the chain of inclusions Manifolds↪FréchetManifolds↪DiffeoligicalSpaces↪SmoothSpaces.

I talk about that on p. 13-14.

Posted by: Urs Schreiber on May 15, 2008 9:47 PM | Permalink | Reply to this

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