Notes for AKT-170317/0:10:47: Difference between revisions
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I'm not sure it was mentioned in the course but g_0 is not just a Lie algebra, it is also a Lie bialgebra. |
I'm not sure it was mentioned in the course but g_0 is not just a Lie algebra, it is also a Lie bialgebra. |
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Translating this to G_0 makes G_0 a Poisson manifold. That means there is a Poisson bracket on the space of functions |
Translating this to G_0 makes G_0 a Poisson manifold. That means there is a Poisson bracket on the space of functions |
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F(G_0). Given such a Poisson bracket {.,.} and a Hamiltonian function |
F(G_0). Given such a Poisson bracket {.,.} and a Hamiltonian function <latex>H:G_0 -> R</latex> is enough to write the equations of motion |
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on G_0. They are df/dt = {f,H} |
on G_0. They are df/dt = {f,H} |
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Revision as of 13:40, 18 March 2017
Here's a way to think about quantum groups: they are integrable quantum mechanical systems that live on Lie groups. To make sense of this let's first upgrade our 4d Lie algebra to a Lie group, call it G_0. I'm not sure it was mentioned in the course but g_0 is not just a Lie algebra, it is also a Lie bialgebra. Translating this to G_0 makes G_0 a Poisson manifold. That means there is a Poisson bracket on the space of functions F(G_0). Given such a Poisson bracket {.,.} and a Hamiltonian function <latex>H:G_0 -> R</latex> is enough to write the equations of motion on G_0. They are df/dt = {f,H}
It's easier to pass to the Lie-group G_0 that corresponds to
