There are lecture notes by Roubtsov-Suchánek on Poisson algebras, where the subject is rendered very algebraically. I believe section 4 has an answer to your question, especially Proposition 4.1., almost by definition.

Suppose that our Poisson structure comes from a non-degenerate closed skew-form $\omega$. The map $\varphi :H\mapsto -X_H$ is a homomorphism of Lie algebras and the center is its kernel (this is where the space of functions gets its structure of a Lie algebra). But this map is nothing more than a composition of the differential of a function with the isomorphism induced by $\omega$, hence the differential of an element of the kernel should be identically zero.

There are lecture notes by Roubtsov-Suchánek on Poisson algebras, where the subject is rendered very algebraically. I believe section 4 has an answer to your question, especially Proposition 4.1., almost by definition.

Suppose that our Poisson structure comes from a non-degenerate closed skew-form $\omega$. The map $\varphi :H\mapsto -X_H$ is an homomorphism of Lie algebras and the center is its kernel (this is where the space of functions gets its structure of a Lie algebra). But this map is nothing more than plugging the differential $dH$ of a function to the isomorphism $T^*X\to TX$ induced by $\omega$, hence the differential of an element of the kernel should be identically zero.

There are lecture notes by Roubtsov-Suchánek on Poisson algebras, where the subject is rendered very algebraically. I believe section 4 has an answer to your question, especially Proposition 4.1., almost by definition.

Suppose that our Poisson structure comes from a non-degenerate closed skew-form $\omega$. The map $\varphi :H\mapsto -X_H$ is a homomorphism of Lie algebras and the center is its kernel (this is where the space of functions gets its structure of a Lie algebra). But this map is nothing more than a composition of the differential of a function with the isomorphism induced by $\omega$, hence the differential should be identically zero.

There are lecture notes by Roubtsov-Suchánek on Poisson algebras, where the subject is rendered very algebraically. I believe section 4 has an answer to your question, especially Proposition 4.1., almost by definition.

Suppose that our Poisson structure comes from a non-degenerate closed skew-form $\omega$. The map $\varphi :H\mapsto -X_H$ is a homomorphism of Lie algebras and the center is its kernel (this is where the space of functions gets its structure of a Lie algebra). But this map is nothing more than a composition of the differential of a function with the isomorphism induced by $\omega$, hence the differential of an element of the kernel should be identically zero.