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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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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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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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This sounds really interesting, I like how g∈𝔤 is reinterpreted as a control, it makes sense because 𝔤 almost completely determines G. But this seems limited to control problems whose state space can be modeled as a Lie group. I wonder how useful this is in practice?
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