Lie Group G
this is a space of reversible things you can do to something that you can do just a little bit of
so like, if you have a cube, you can do rotations to it, even lil rotations: boom! that's a Lie Group right there SO(3)
in this case, G are actions you can do to P...
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Poisson Manifold P
oh boy, what the heck is this?
this is like a space of things you can do Fourier transforms to (like symplectic manifolds, but more badass)
in particular it's a space which has a special bivector field on it which has to commute with itself
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but wait, doesn't everything commute with itself?
well, it's true for vectors, but no longer true for bivectors
if our bivector is a ^ b, then the commutator for itself will be -2 [a, b] ^ a ^ b
this ends up boiling down to the Jacobi identity, but i think this is more natural
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i'm still struggling to intuit this, so i'd love to hear other ideas on what exactly this all means
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oh, but anyway, the formula above geometrically means that we can think of a bivector commuting with itself as meaning that the commutator of the vectors that make up the bivector need to lie within the bivector
so it can be curved, but it has to be curved "along" the bivector
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(btw, this isn't actual "curvature" since the Poisson bivector is not a metric, but it is analogous to curvature since that's what the commutator of a metric is)
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Lie Algebra 𝖌
these are the smallest possible actions from G
i like to imagine this like the stasis control from Breath of the Wild
gfycat.com/repentantlasta
for the Lie Group SO(3), the Lie Algebra is called so(3), and is just R^3: a slight twist in xy, in yz, and zx planes
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but what makes Lie Algebras special is the commutator bracket
try rotating an object a bit in the xy plane, then the yz plane, then back the same amount in the xy plane, and then back the same amount in the yz plane (i.e. [xy, yz])
it *doesn't* end up how it started!
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Replying to
You’re a pure math researcher right? Have you ever run into the control engineerin side of this? Control on Lie groups was a big idea in robotics in particular when I was in grad school, especially for nonholonomic constraint mechanisms.
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a long time ago i was a grad student (dropped out), but i do this for fun now!
i don't think i've seen that in particular, but the idea makes a lot of sense!
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Starting point in case you’re interested. Turned out to be too hard for me 🙃
citeseerx.ist.psu.edu/viewdoc/downlo

