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Is there a symplectic structure on $M_{2n}(\mathbb{R})$, not necessarily with constant coefficients, such that the space of smooth invariant functions, those smooth functions $f:M_{2n}(\mathbb{R})\to \mathbb{R}$ with $f(AB)=f(BA)\;\;\;\; \forall A,B \in M_{2n}(\mathbb{R})$, would be closed under the Poisson bracket ?

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If I understood correctly a rephrasing of question is requiring a symplectic structure such that the set of commutators $AB-BA$ is contained inside a coisotropic submanifold. It shouldn't be difficult to find a symplectic structure on the space of matrices such that the linear subspace $\mathfrak{sl}_n$ (which contains all commutators, being linearly generated by it) is coisotropic.

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  • $\begingroup$ thank you for your answer. Why are commutators related to my question while an invariant polynomial is not necessarily a linear map? $\endgroup$ – Ali Taghavi Jul 23 '16 at 22:28

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