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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)(g,A) to Lie groupoids C(g,A)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 ∞\infty-groups and action Lie ∞\infty-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(−)\Pi_1(-) of the smooth classifying space S(CE(g,A))S(\mathrm{CE}(g,A)) of (g,A)(g,A)-valued differential forms: C(g,A)=Π 1(S(CE(g,A))). C(g,A) = \Pi_1(S(CE(g,A))) \,.

Posted at May 7, 2008 8:41 AM UTC

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5 Comments & 4 Trackbacks

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 ∞L_\infty-algebroid integrates to an ∞\infty-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. Manifolds \hookrightarrow FréchetManifolds \hookrightarrow DiffeoligicalSpaces \hookrightarrow SmoothSpaces \,.

I talk about that on p. 13-14.

Posted by: Urs Schreiber on May 15, 2008 9:47 PM | Permalink | Reply to this
Read the post A Groupoid Approach to Quantization
Weblog: The n-Category Café
Excerpt: On Eli Hawkins' groupoid version of geometric quantization.
Tracked: June 12, 2008 5:48 PM
Read the post Eli Hawkins on Geometric Quantization, I
Weblog: The n-Category Café
Excerpt: Some basics and some aspects of geometric quantization. With an emphasis on the geometric quantization of duals of Lie algebras and duals of Lie algebroids.
Tracked: June 20, 2008 5:07 PM
Read the post Eli Hawkins on Geometric Quantization, II
Weblog: The n-Category Café
Excerpt: Eli Hawkins explains his method of getting a quantum algebra from the convolution algebra of sections on a symplectic groupoid.
Tracked: June 27, 2008 5:43 PM

Re: Integrability of Lie Brackets

This is now being linked to from the entry Lie theory at the nn-Lab.

Posted by: Urs Schreiber on December 1, 2008 9:00 PM | Permalink | Reply to this

Re: Integrability of Lie Brackets

the link n-catLab.org doesn’t

Posted by: jim stasheff on December 2, 2008 12:21 AM | Permalink | Reply to this

Re: Integrability of Lie Brackets

the link n-catLab.org doesn’t

The link is ncatlab.org, without the hyphen.

But it can also still happen, it seems, that the site is down for a bit. While it seems to run stably now, I had to reboot the server yesterday again after it became irresponsive (but without the former 503 error, so it was somehting else).

So if the above link doesn’t work for you, please wait a bit and try again later.

Posted by: Urs Schreiber on December 2, 2008 8:49 AM | Permalink | Reply to this

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