A Poisson ring
is a commutative ring
on which a binary operation
, known as the Poisson bracket is defined. This operation must satisfy the following identities:
for all scalars
Because of properties 2 and 3, for each
, the operation
defined as
is a derivation. If the set
generates the set of derivations of
, we say that
is non-degenerate.
It can be shown that, if
is non-degenerate and is isomorphic as a commutative ring to the algebra of smooth functions
on a manifold
, then
must be a symplectic manifold and
is the Poisson bracket defined by the symplectic form.
Many important operations and results of symplectic geometry and Hamiltonian mechanics may be formulated in terms of the Poisson bracket and, hence, apply to Poisson algebras as well. This observation is important in studying the classical limit of quantum mechanics -- the non-commutative algebra of operators on a Hilbert space has the Poisson algebra of functions on a symplectic manifold as a singular limit and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.
In addition to their use in mechanics, Poisson algebras are also used in the study of Lie groups.
As of this snapshot date, this entry was owned by rspuzio.