Poisson vertex algebra

Suppose $vir_{c}= \operatorname{span}\langle L_{-2}v_{c},L_{-3}v_{c},....\rangle$ is a vector space spanned by Virasoro algebra. Then we have a symmetric algebra $Sym(vir_{c})$. For this symmetric algebra we can endow it with a commutative vertex algebra structure by $Y_{+}(a,z)=e^{zL_{-1}}a$. But in order to make it into a poisson vertex algebra, we have to define $Y_{-}$ on symmetric algebra to give it a vertex Lie algebra structure. I know $Y_{-}(L_{-2}v_{c}，z)=\Sigma_{n\geq -1}L_{n}z^{-n-2}$ where $L_{n}$ acts as derivation via commutation relation of virasora algebra. But I don't know how to define a general case, such as $Y_{-}(L_{-3}v_{c}.L_{-2}v_{c},z)$. All the notations follow from the book vertex algebras and algebraic curves by Edward Frenkel and David Ben-Zvi chapter 16.

This is an exercise problem and it is more proper to ask it on stack exchange. Your problem is that you do not know how to evaluate $Y_{-}(a\cdot b,z)$ for arbitrary $a,b$ in the Poisson vertex algebra (PVA) as it is not told by the vertex Lie algebra structure (Here $\cdot$ denotes the commutative product, which is $\circ$ in your ref.). The answer is that you take another element $c$ in the PVA and consider $Y_{-}(a\cdot b,z) c$. Using skew -symmetry (P.268 Def. 16.1.1 in your ref.) and derivation property of PVA (P.271, Def. 16.2.1 in your ref.) you will get the answer quickly. I suggest that you can see the papers and notes by Kac and his collaborators on PVA if you feel your ref. does not tell you enough. For example the following lecture notes by Kac (or its published Springer version):