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
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
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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
hmm well for an action lie algebroid coming from an action by G it should just be the lie bracket on g, that case at least reduces to the usual lie algebra story
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