Talk:PlanetPhysics/Poisson Ring
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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Poisson ring %%% Primary Category Code: 45.20.-d %%% Filename: PoissonRing.tex %%% Version: 3 %%% Owner: rspuzio %%% Author(s): bci1, rspuzio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}
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A \emph{Poisson ring} $A$ is a \htmladdnormallink{commutative ring}{http://planetphysics.us/encyclopedia/PAdicMeasure.html} on which a binary \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} $[,]$, known as the Poisson bracket is defined. This operation must satisfy the following \htmladdnormallink{identities}{http://planetphysics.us/encyclopedia/Identity2.html}:
\begin{enumerate} \item $[f,g] = -[g,f]$ \item $[f + g, h] = [f,h] + [g,h]$ \item $[fg,h] = f[g,h] + g[f,h]$ \item $[f,[g,h]] + [g,[h,f]] + [h,[f,g]] = 0$ \end{enumerate} If, in addition, $A$ is an algebra over a \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}, then we call $A$ a \emph{Poisson algebra}. In this case, we may wish to add the extra requirement $$[sf,g] = s[f,g]$$ for all \htmladdnormallink{scalars}{http://planetphysics.us/encyclopedia/Vectors.html} $s$.
Because of properties 2 and 3, for each $g \in A$, the operation $ad_g$ defined as $ad_g(f) = [f,g]$ is a derivation. If the set $\{ ad_g | g \in A \}$ generates the set of derivations of $A$, we say that $A$ is \emph{non-degenerate}.
It can be shown that, if $A$ is non-degenerate and is isomorphic as a commutative ring to the algebra of smooth \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} on a \htmladdnormallink{manifold}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} $M$, then $M$ must be a symplectic manifold and $[,]$ is the Poisson bracket defined by the symplectic form.
Many important operations and results of symplectic geometry and \htmladdnormallink{Hamiltonian}{http://planetphysics.us/encyclopedia/Hamiltonian2.html} \htmladdnormallink{mechanics}{http://planetphysics.us/encyclopedia/Mechanics.html} 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 \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html} --- the \htmladdnormallink{non-commutative algebra}{http://planetphysics.us/encyclopedia/MoritaInvariant.html} of \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} on a \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} 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 \htmladdnormallink{Lie groups}{http://planetphysics.us/encyclopedia/BilinearMap.html}.
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