Poisson bracket and $L_\infty$ algebras

We describe the Poisson bracket of a Lagrangian field theory expressed in the framework of $L_\infty$ algebras.

We show that the recently proposed symplectic structure implies that the associated Poisson bracket can be computed through the Peierls formula.

We consider Poisson brackets in $p$-adic string theory, where interesting complications arise.

In addition we give an elegant interpretation of the inverse relation between the Poisson bracket and symplectic structure in the language of homological algebra, extending some ideas in the mathematical physics literature.