Lie algebroid structures on C^∞(M, g) induce Poisson brackets on C^∞(M × g*) that include but strictly exceed the Darboux and Lie-Poisson cases, with computational examples and central extensions studied.
Hirsch, (1976) Differential Topology, Graduate Texts in Mathematics 33, Springer Verlag, New York
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On Poisson structures arising from a Lie group action
Lie algebroid structures on C^∞(M, g) induce Poisson brackets on C^∞(M × g*) that include but strictly exceed the Darboux and Lie-Poisson cases, with computational examples and central extensions studied.