%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: Hamiltonian algebroid
%%% Primary Category Code: 03.
%%% Filename: HamiltonianAlgebroid2.tex
%%% Version: 9
%%% Owner: bci1
%%% Author(s): bci1
%%% 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}
\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}
\usepackage{html}
% this is the default PlanetPhysics preamble. as your
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% define commands here
\newcommand*{\abs}[1]{\left\lvert #1\right\rvert}
\newtheorem{prop}{Proposition}
\newtheorem{thm}{Theorem}
\newtheorem{ex}{Example}
\newcommand{\real}{\mathbb{R}}
\newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}}
\usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym}
\usepackage{xypic}
\usepackage[mathscr]{eucal}
\theoremstyle{plain}
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]
%\theoremstyle{remark}
\newtheorem{remark}{Remark}[section]
\newtheorem*{notation}{Notation}
\newtheorem*{claim}{Claim}
\renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@
}}}
\numberwithin{equation}{section}
\newcommand{\Ad}{{\rm Ad}}
\newcommand{\Aut}{{\rm Aut}}
\newcommand{\Cl}{{\rm Cl}}
\newcommand{\Co}{{\rm Co}}
\newcommand{\DES}{{\rm DES}}
\newcommand{\Diff}{{\rm Diff}}
\newcommand{\Dom}{{\rm Dom}}
\newcommand{\Hol}{{\rm Hol}}
\newcommand{\Mon}{{\rm Mon}}
\newcommand{\Hom}{{\rm Hom}}
\newcommand{\Ker}{{\rm Ker}}
\newcommand{\Ind}{{\rm Ind}}
\newcommand{\IM}{{\rm Im}}
\newcommand{\Is}{{\rm Is}}
\newcommand{\ID}{{\rm id}}
\newcommand{\GL}{{\rm GL}}
\newcommand{\Iso}{{\rm Iso}}
\newcommand{\Sem}{{\rm Sem}}
\newcommand{\St}{{\rm St}}
\newcommand{\Sym}{{\rm Sym}}
\newcommand{\SU}{{\rm SU}}
\newcommand{\Tor}{{\rm Tor}}
\newcommand{\U}{{\rm U}}
\newcommand{\A}{\mathcal A}
\newcommand{\Ce}{\mathcal C}
\newcommand{\D}{\mathcal D}
\newcommand{\E}{\mathcal E}
\newcommand{\F}{\mathcal F}
\newcommand{\G}{\mathcal G}
\newcommand{\Q}{\mathcal Q}
\newcommand{\R}{\mathcal R}
\newcommand{\cS}{\mathcal S}
\newcommand{\cU}{\mathcal U}
\newcommand{\W}{\mathcal W}
\newcommand{\bA}{\mathbb{A}}
\newcommand{\bB}{\mathbb{B}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\bD}{\mathbb{D}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bG}{\mathbb{G}}
\newcommand{\bK}{\mathbb{K}}
\newcommand{\bM}{\mathbb{M}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bO}{\mathbb{O}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bV}{\mathbb{V}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\bfE}{\mathbf{E}}
\newcommand{\bfX}{\mathbf{X}}
\newcommand{\bfY}{\mathbf{Y}}
\newcommand{\bfZ}{\mathbf{Z}}
\renewcommand{\O}{\Omega}
\renewcommand{\o}{\omega}
\newcommand{\vp}{\varphi}
\newcommand{\vep}{\varepsilon}
\newcommand{\diag}{{\rm diag}}
\newcommand{\grp}{{\mathbb G}}
\newcommand{\dgrp}{{\mathbb D}}
\newcommand{\desp}{{\mathbb D^{\rm{es}}}}
\newcommand{\Geod}{{\rm Geod}}
\newcommand{\geod}{{\rm geod}}
\newcommand{\hgr}{{\mathbb H}}
\newcommand{\mgr}{{\mathbb M}}
\newcommand{\ob}{{\rm Ob}}
\newcommand{\obg}{{\rm Ob(\mathbb G)}}
\newcommand{\obgp}{{\rm Ob(\mathbb G')}}
\newcommand{\obh}{{\rm Ob(\mathbb H)}}
\newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}}
\newcommand{\ghomotop}{{\rho_2^{\square}}}
\newcommand{\gcalp}{{\mathbb G(\mathcal P)}}
\newcommand{\rf}{{R_{\mathcal F}}}
\newcommand{\glob}{{\rm glob}}
\newcommand{\loc}{{\rm loc}}
\newcommand{\TOP}{{\rm TOP}}
\newcommand{\wti}{\widetilde}
\newcommand{\what}{\widehat}
\renewcommand{\a}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\de}{\delta}
\newcommand{\del}{\partial}
\newcommand{\ka}{\kappa}
\newcommand{\si}{\sigma}
\newcommand{\ta}{\tau}
\newcommand{\lra}{{\longrightarrow}}
\newcommand{\ra}{{\rightarrow}}
\newcommand{\rat}{{\rightarrowtail}}
\newcommand{\oset}[1]{\overset {#1}{\ra}}
\newcommand{\osetl}[1]{\overset {#1}{\lra}}
\newcommand{\hr}{{\hookrightarrow}}
\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}
