Talk:PlanetPhysics/Hamiltonian Algebroid 2

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%%% Primary Title: Hamiltonian algebroid
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%%% Filename: HamiltonianAlgebroid2.tex
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\begin{document}

 \subsection{Introduction}
\emph{Hamiltonian algebroids} are generalizations of the \htmladdnormallink{Lie algebras}{http://planetphysics.us/encyclopedia/BilinearMap.html} of canonical transformations.


\begin{definition}
Let $X$ and $Y$ be two \htmladdnormallink{vector fields}{http://planetphysics.us/encyclopedia/NeutrinoRestMass.html} on a smooth \htmladdnormallink{manifold}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} $M$, represented here as \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} acting on \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html}.
Their \htmladdnormallink{commutator}{http://planetphysics.us/encyclopedia/Commutator.html}, or Lie bracket, $L$, is :
\begin{align*}
[X,Y](f)=X(Y(f))-Y(X(f)).
\end {align*}


Moreover, consider the classical configuration space $Q = \bR^3$ of a classical, mechanical \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}, or \htmladdnormallink{particle}{http://planetphysics.us/encyclopedia/Particle.html} whose phase space is the cotangent bundle $T^* \bR^3 \cong \bR^6$, for which the space of (classical)
\htmladdnormallink{observables}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} is taken to be the real \htmladdnormallink{vector space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} of smooth functions on $M$, and with T being an element
of a \htmladdnormallink{Jordan-Lie (Poisson) algebra}{http://planetphysics.us/encyclopedia/JordanBanachAndJordanLieAlgebras.html} whose definition is also recalled next. Thus, one defines as in classical \htmladdnormallink{dynamics}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} the \emph{\htmladdnormallink{Poisson algebra}{http://planetphysics.us/encyclopedia/PoissonRing.html}} as a Jordan algebra in which $\circ$ is associative. We recall that one needs to consider first a \emph{specific algebra} (defined as a vector space $E$ over a ground \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} (typically $\bR$ or $\bC$)) equipped with a bilinear and distributive multiplication $\circ$~. Then one defines a \emph{Jordan algebra} (over $\bR$), as a a specific algebra over $\bR$ for which:


$ \begin{aligned} S \circ T &= T \circ S~, \\ S \circ (T \circ S^2) &= (S \circ T) \circ S^2 ,
\end{aligned},$

for all elements $S, T$ of this algebra.

Then, the usual \htmladdnormallink{algebraic}{http://planetphysics.us/encyclopedia/CoIntersections.html} \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} automorphism, \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}, etc.) apply to a
\htmladdnormallink{Jordan-Lie (Poisson) algebra}{http://planetphysics.us/encyclopedia/JordanBanachAndJordanLieAlgebras.html} defined as a real vector space $\mathfrak A_{\bR}$ together with a \emph{Jordan product} $\circ$ and \emph{Poisson bracket}

$\{~,~\}$, satisfying~:

\begin{itemize}
\item[1.] for all $S, T \in \mathfrak A_{\bR},$

$\begin{aligned} S \circ T &= T \circ S \\ \{S, T \} &= - \{T,
S\} \end{aligned}$

\item[2.] the \emph{Leibniz rule} holds

$ \{S, T \circ W \} = \{S, T\} \circ W + T \circ \{S, W\}$
for all $S, T, W \in \mathfrak A_{\bR}$, along with


\item[3.]

the \emph{Jacobi \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html}}~:

$$ \{S, \{T, W \}\} = \{\{S,T \}, W\} + \{T, \{S, W \}\}$$


\item[4.]

for some $\hslash^2 \in \bR$, there is the \emph{associator identity} ~:

$$(S \circ T) \circ W - S \circ (T \circ W) = \frac{1}{4} \hslash^2 \{\{S, W \}, T \}~.$$



\end{itemize}


Thus, the canonical transformations of the Poisson sigma model phase space specified by the \htmladdnormallink{Jordan-Lie (Poisson) algebra}{http://planetphysics.us/encyclopedia/JordanBanachAndJordanLieAlgebras.html} (also Poisson algebra), which is determined by both the Poisson bracket and the \emph{Jordan product} $\circ$, define a \emph{Hamiltonian algebroid} with the Lie brackets $L$ related to such a Poisson structure on the target space.
\end{definition}

\end{document}