does anyone wanna explain lie algebroids to me
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it’s like uhhh you take a lie groupoid and differentiate it right
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yeah okay ummm...
what? does that means exactly??
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okay so a lie groupoid is like a smooth manifold together with a bunch of “smooth isomorphisms” between points right? (thinking about a smooth action of a lie group helps here maybe)
you differentiate that and get infinitesimal isomorphisms between infinitesimally close points
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this is the vector bundle + the anchor map in the anchor map definition; the vector bundle is “infinitesimal isomorphisms” and the map to the tangent bundle tells you which infinitesimally close points are being identified
then the lie bracket encodes composition
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so e.g. the tangent lie algebroid, where the bundle is the tangent bundle itself, with anchor the identity, is trying to identify *all* infinitesimally nearby points; it formally exponentiates to the “path lie groupoid” PX given by all smooth paths
hmm maybe it’s easier to think of it as exponentiating to X x X together with the two projections to X, which identifies every pair of points but uniquely
actually the path lie groupoid is probably too big welp