Let $V$ be a finite dimensional vector space and $S(V)$ the corresponding symmetric algebra. Suppose that we have a Poisson bracket $\lambda = \{,\}: S(V) \otimes S(V) \to S(V)$. Let $V^*$ be the dual vector space of $V$. Is there a natural Poisson bracket on $S(V^*)$ which is related to the Poisson bracket $\lambda$ on $S(V)$? Thank you very much.

## 1 Answer

Under the natural interpretation of the term "natural", the answer is negative. For example, consider a linear bracket corresponding to a Lie algebra that does not admit an invariant bilinear form (e.g., 3-dimensional Heisenberg Lie algebra). The dual vector space has no "natural" Poisson bracket.

The answer is positive in the important special case of *nondegenerate* Poisson bracket, because in this case $V$ and its dual are canonically isomorphic by means of the corresponding symplectic structure.

gradedPoisson structure on the exterior algebra $\Lambda(V^*)$ rather than on $S(V^*)$ (and the two Poisson enveloping algebras are Koszul duals). $\endgroup$