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ADITTYA CHAUDHURI (Feb 11 2022 at 21:15):Are there any results on decomposing a Lie groupoid(appropriately) into sub Lie groupoids? Like in the case of manifolds we have the notion of foliation.
John Baez (Feb 12 2022 at 00:20):I feel I should, but I don't know any results like that!
ADITTYA CHAUDHURI (Feb 12 2022 at 07:29):John Baez said:
I feel I should, but I don't know any results like that!
Sir, can you suggest any literature in that direction? (Even if they are not results!)
John Baez (Feb 12 2022 at 16:55):I don't know any.
ADITTYA CHAUDHURI (Feb 12 2022 at 20:45):Ok Sir.
David Michael Roberts (Feb 12 2022 at 22:45):I can give two examples, as perhaps that illustrates the general idea: cut up the manifold of objects into its orbits. If the groupoid is proper, then we know this is I think a Whitney stratification. By construction it is actually a decomposition of the groupoid, not just the objects.
Another example (again assume proper, for safety) is the decomposition into stabiliser types, which is a priori coarser than the previous one. Two objects x and y (points in the object manifold) have the same stabiliser type if the representations on the tangent spaces at x and y induced by local slices transverse to the orbits are isomorphic. Every object in an orbit has the same stabiliser type, but it could be that objects in different orbits also do. Then each piece of the decomposition of the manifold of objects is a union of orbits, and so again gives a decomposition of the whole groupoid.
I'm not sure about a decomposition into subgroupoids that's finer than the orbit decomposition. The only example I can think of is into the fibres of a groupoid extension, but I haven't checked the details (and so it might not even work!), but I don't think this is as subgroupoids.
ADITTYA CHAUDHURI (Feb 13 2022 at 04:56):David Michael Roberts said:
I can give two examples, as perhaps that illustrates the general idea: cut up the manifold of objects into its orbits. If the groupoid is proper, then we know this is I think a Whitney stratification. By construction it is actually a decomposition of the groupoid, not just the objects.
Another example (again assume proper, for safety) is the decomposition into stabiliser types, which is a priori coarser than the previous one. Two objects x and y (points in the object manifold) have the same stabiliser type if the representations on the tangent spaces at x and y induced by local slices transverse to the orbits are isomorphic. Every object in an orbit has the same stabiliser type, but it could be that objects in different orbits also do. Then each piece of the decomposition of the manifold of objects is a union of orbits, and so again gives a decomposition of the whole groupoid.
I'm not sure about a decomposition into subgroupoids that's finer than the orbit decomposition. The only example I can think of is into the fibres of a groupoid extension, but I haven't checked the details (and so it might not even work!), but I don't think this is as subgroupoids.
Thanks a lot for the examples.
ADITTYA CHAUDHURI (Feb 13 2022 at 05:44):@David Michael Roberts "If the groupoid is proper, then we know this is I think a Whitney stratification. By construction it is actually a decomposition of the groupoid, not just the objects" . Can you please suggest any literature where this example is mentioned?
David Michael Roberts (Feb 13 2022 at 06:40):I think searching whitney stratification proper lie groupoid will find you some
ADITTYA CHAUDHURI (Feb 13 2022 at 07:58):David Michael Roberts said:
I think searching
whitney stratification proper lie groupoidwill find you some
Ok, I will do that. Thanks!
David Michael Roberts (Feb 13 2022 at 09:43):No worries! I should say that since slice theorems are involved, that's a good one to add. People to look for are Pflaum, Posthuma, Tang, Crainic. But there are more.
ADITTYA CHAUDHURI (Feb 13 2022 at 13:41):David Michael Roberts said:
No worries! I should say that since slice theorems are involved, that's a good one to add. People to look for are Pflaum, Posthuma, Tang, Crainic. But there are more.
Thanks a lot!