(Commutative) Poisson algebras are clearly very interesting, so it would be nice to have ways of constructing examples. We know that is a Poisson algebra with bracket uniquely defined by ; this describes a classical particle in one dimension, and is the classical limit of a quantum particle in one dimension (essentially the Weyl algebra).
More generally, if are Poisson algebras, then the tensor product can be given a Poisson bracket given by extending
linearly. At least when are unital, this Poisson algebra is the universal Poisson algebra with Poisson maps from such that the images of elements of Poisson-commute with the images of elements of . In particular, it follows that is a Poisson algebra with the bracket
This describes a classical particle in dimensions, or different classical particles in one dimension, and it is the classical limit of a quantum particle in dimensions, or different quantum particles in one dimension.
Today we’ll discuss the question of how one might go about constructing Poisson brackets more generally.